Foliations in deformation spaces of local G-shtukas

Urs Hartl and Eva Viehmann

Abstract We study local G-shtukas with level structure over a base scheme whose Newton polygons are constant on the base. We show that after a finite base change and after passing to an étale covering, such a local G-shtuka is isogenous to a completely slope divisible one, generalizing corresponding results for p-divisible groups by Oort and Zink. As an application we establish a product structure up to finite surjective morphism on the closed Newton stratum of the universal deformation of a local G-shtuka, similarly to Oort’s foliations for p-divisible groups and abelian varieties. This also yields bounds on the dimensions of affine Deligne-Lusztig varieties and proves equidimensionality of affine Deligne-Lusztig varieties in the affine Grassmannian. Mathematics Subject Classification (2000): 20G25 (11G09, 14L05, 14M15)

1 Introduction

We denote by 𝔽_{q} the finite field with q elements and by k an algebraically closed field which is also an 𝔽_{q}-algebra.

Let G be a split connected reductive group over 𝔽_{q}. Local G-shtukas are analogs over the function field 𝔽_{q}((z)) of p-divisible groups with extra structure. To define them, let LG be the loop group of G see [Fa, Definition 1]. I.e. LG is the ind-scheme over 𝔽_{q} representing the sheaf of groups for the fpqc-topology whose sections for an 𝔽_{q}-algebra R are given by LG(Spec R) = G(R[[z]][\frac{1}{z}]). We also use the analogous notion for example for parabolic subgroups of G. Let K_{0} be the infinite dimensional affine group scheme over 𝔽_{q} with K_{0}(Spec R) = G(R[[z]]). In the natural way K_{0} can be viewed as a subsheaf of LG. Every K_{0}-torsor over an 𝔽_{q}-scheme S for the fpqc-topology is already a K_{0}-torsor for the étale topology by [HV, Proposition 2.1]. For such a K_{0}-torsor G on S let LG be the associated LG-torsor and σ^{∗}LG the pullback of LG under the q-Frobenius morphism Frob_{q} : S → S. Then a local G-shtuka over S is a pair \underline{G} = (G, φ) consisting of a K_{0}-torsor G on S and an isomorphism of LG-torsors φ : σ^{∗}LG ^{∼} ⟶ LG. Local G-shtukas over k can also be described as follows. There exists a trivialization G ≅ (K_{0})_{k} and with respect to such a trivialization φ corresponds to an element b ∈ LG(k). A change of the trivialization replaces b by g^{−1}bσ^{∗}(g) for g ∈ K_{0}(k), where σ^{∗} is the endomorphism of K_{0} and of LG induced by Frob_{q}. The special case of local GL_{n}-shtukas is a direct analog of p-divisible groups in this context of function fields [Ha]. For G = GSp_{2n} one obtains the analog of polarized p-divisible groups. For a definition of local G-shtukas and their moduli over more general schemes see [HV]. In contrast to [HV] we here only consider local G-shtukas over base schemes S over 𝔽_{q} = 𝔽_{q}[[ζ]]/(ζ) instead of over 𝔽_{q}[[ζ]]. The restriction to base schemes with ζ = 0 corresponds to p-divisible groups over 𝔽_{p}-schemes instead of over ℤ_{p}-schemes. Whenever it does not require additional effort we also consider more generally local G-shtukas with a level structure by certain subgroups K of K_{0}, the most interesting cases being K = K_{0} (in which case one obtains the definition above) or K = I, an Iwahori subgroup of K_{0}. For simplicity we only consider the case K = K_{0} in the introduction.

Let B ⊃ T be a Borel subgroup and a maximal split torus of G. Recall that a coweight λ ∈ X_{∗}(T) is called dominant if ⟨λ, α⟩ ≥ 0 for every positive root α of G. To a local G-shtuka \underline{G} over an algebraically closed field k we assign its Hodge point or Hodge polygon, i.e. the unique dominant coweight μ = μ\underline{G} ∈ X_{∗}(T) such that the element b associated with \underline{G} is in the double coset K_{0}(k)z^{μ}K_{0}(k) where z^{μ} denotes the image of z under μ : 𝔾_{m} → T. For \underline{G} over a scheme S we consider the function μ\underline{G} assigning to each geometric point s of S the Hodge point μ\underline{G}(s) of \underline{G}_{s}. It has the property that its image \overline{μ}\underline{G} in π_{1}(G) = X_{∗}(T)/(coroot lattice) is locally constant on S (see [HV, Proposition 3.4]). If S is reduced we say that \underline{G} is bounded by some dominant μ ∈ X_{∗}(T) if μ\underline{G}(s) ⪯ μ for all s ∈ S. Here ⪯ is the Bruhat ordering on the set of dominant coweights of G, i.e. μ_{1} ⪯ μ_{2} if and only if the difference μ_{2} − μ_{1} is a non-negative integral linear combination of simple coroots. There is also a notion of boundedness for local G-shtukas over non-reduced schemes, compare Definition 3.5 or [HV, Definition 3.5].

A second important invariant of a local G-shtuka over an algebraically closed field is its isogeny class. It corresponds to the σ-conjugacy class [b] = {g^{−1}bσ^{∗}(g) : g ∈ LG(k)} of the associated element b ∈ LG(k). In [Ko], Kottwitz classifies the set B(G) of σ-conjugacy classes of elements b ∈ LG(k) by two invariants. One of them is κ(b) := \overline{μ}_{(K_{k},bσ^{∗})}. Note that it can also be defined directly from b via the Kottwitz map κ : LG(k) → π_{1}(G). Especially, it only depends on [b]. The other invariant is the Newton point or Newton polygon ν_{b} ∈ X_{∗}(T)_{ℚ} of b. For G = GL_{n} this is the usual Newton polygon of the σ-linear map bσ^{∗}, for general G it is defined by requiring that it be functorial in G. The two invariants have the same image in π_{1}(G)_{ℚ}. For \underline{G} = (K_{k}, bσ^{∗}) we have ν\underline{G} ⪯ μ\underline{G} as elements of X_{∗}(T)_{ℚ} (see [HV, Proposition 7.2]). The Newton point of a local G-shtuka \underline{G} over a scheme S is defined as the function assigning to each k-valued point s of S the Newton point ν = ν\underline{G}(s) of \underline{G}_{s}. Over schemes, and already over general fields, two local G-shtukas whose Newton points coincide are not necessarily isogenous. Let (ν, \overline{μ}) be the pair of invariants associated with [b] for some b ∈ LG(k). Let

N_{ν} = \{s ∈ S(k) : ν\underline{G}(s) = ν, \overline{μ}_{\underline{G}_{s}} = \overline{μ}\}.

By [HV, Theorem 7.3] (see also [RR, Theorem 3.6]) this is the set of points of a locally closed subscheme of S which is open in the analogously defined subscheme where we replace equality of Newton points by the condition ν\underline{G}(s) ⪯ ν. By abuse of notation we do not include \overline{μ} in the notation (although the stratum depends on \overline{μ}) and call N_{ν} the Newton stratum associated with ν. The reason for this is that \overline{μ} is locally constant on S, so the additional datum \overline{μ} only singles out specific connected components.

We now describe the results obtained in this article. Let \underline{G} be a local G-shtuka over a noetherian scheme S whose Newton point is constant. Let M be the centralizer of the Newton point. It is the Levi component containing T of a standard parabolic subgroup P of G, i.e. of a parabolic subgroup containing B. Let N be the unipotent radical of P. Let \overline{P} be the opposite parabolic. We say that \underline{G} has a slope filtration if there exists a local \overline{P} -shtuka \underline{P̄} = (P̄, φ) over S such that \underline{G} ≅ i_{∗}\underline{P̄} where i : \overline{P} ↪ G is the inclusion morphism, and étale locally on S the Frobenius φ is represented by an element b ∈ L\overline{P} (S) with decomposition b = m\overline{n} for some m with M-dominant Newton point equal to ν.

In Section 4 we consider a stronger condition than having a slope filtration, namely to be completely slope divisible. In addition to having a slope filtration it requires certain integrality conditions on the elements m and \overline{n}. It is related to the condition for p-divisible groups to be completely slope divisible defined by Zink in [Zi, Definition 10], compare Section 5. Our main result on slope filtrations or rather on completely slope divisible local G-shtukas is the following theorem.

Theorem 4.14. Let S be an integral noetherian 𝔽_{q}-scheme and let g ∈ LG(S) be such that the σ-conjugacy classes of g_{s} in the geometric points s of S all coincide. Let x be a P -fundamental alcove associated with this σ-conjugacy class as in Remark 4.2 (a). Then there are morphisms S̃ ⟶^{β} S′ ⟶^{α} S with S′ integral, α finite surjective, and β an étale covering, and a bounded element h̃ ∈ LG(S̃) satisfying h̃^{−1}gS̃σ^{∗}(h̃) ∈ I(S̃)xI(S̃).

Here I denotes the standard Iwahori subgroup scheme of K_{0}. This theorem together with Proposition 4.8 and Corollary 4.9 implies that if \underline{G} is a local G-shtuka over a scheme S whose quasi-isogeny class is constant in all geometric points of S, then after a suitable finite base change and after passing to an étale covering \underline{G} is isogenous to a local G-shtuka which is completely slope divisible. This is an analog of [Zi, Theorem 7] and [OZ, Lemma 2.4]. The proof is also similar to the one given by Zink respectively Oort and Zink, however there is one ingredient [Zi, Lemma 9] which does not have an immediate generalization from GL_{n} to general reductive groups, so we replace it by a different argument here.

We then use this result to introduce and study the analogs of Oort’s [Oo] foliations and central leaves in our situation. A description of central leaves for a p-divisible group whose Newton polygon has just two slopes which uses slope filtrations can be found in [Ch]. Let 𝔽/𝔽_{q} be a field extension, S a noetherian 𝔽-scheme, and let \underline{G} be a bounded local G-shtuka over S. Let \underline{𝔾} be a local G-shtuka over 𝔽. We consider the subset of S given by

C_{\underline{𝔾},S} := \{ s ∈ S : \underline{𝔾}_{k} ≅ \underline{G}_{s} ⊗_{k(s)} k over an algebraically closed extension k/k(s) \}

Here and throughout the article k(s) denotes the residue field of s. In Corollary 6.7 we show that this defines a reduced locally closed subscheme which is closed in the Newton stratum of \underline{𝔾}. Any geometrically irreducible component of C_{\underline{𝔾},S} with the induced reduced subscheme structure is called a central leaf corresponding to \underline{𝔾} in S.

In this way we do not only obtain a decomposition of each Newton stratum into a disjoint union of (in general infinitely many) closed subschemes, but in the special case of the Newton stratum of the universal deformation of a local G-shtuka bounded by some μ the leaves can be arranged in a continuous family. To formulate the corresponding result recall the following definition. The closed affine Deligne-Lusztig variety associated with b ∈ LG(k) and a dominant μ ∈ X_{∗}(T) is the reduced closed subscheme X_{⪯μ}(b) of the affine Grassmannian LG/K_{0} with

X_{⪯μ}(b)(k) = {g ∈ LG(k)/K_{0}(k) : g^{−1}bσ^{∗}(g) ∈ K_{0}(k)z^{μ′}K_{0}(k) for some μ′ ⪯ μ}.

It is the underlying topological space of a Rapoport-Zink space for local G-shtukas; see [HV, Theorem 6.3]. There are also variants where K_{0} is replaced by an Iwahori subgroup or where we require μ′ = μ (see Section 6). In Theorem 6.5 we prove (a slight generalization of) the following theorem.

Theorem. Let \underline{𝔾} = (K_{0,k}, bσ^{∗}) be a local G-shtuka over an algebraically closed field k bounded by a dominant μ ∈ X_{∗}(T) and with Newton point ν. Let N_{ν} be the Newton stratum of [b] in the universal deformation space of \underline{𝔾} bounded by μ. Let x be a P -fundamental alcove associated with [b]. Then there is a reduced scheme S and a finite surjective morphism S ↠ N_{ν} which factorizes into finite surjective morphisms S → X_{⪯μ}(b)^{∧} ˆ×_{k} I^{∧} → N_{ν}. Here X_{⪯μ}(b)^{∧} denotes the completion of X_{⪯μ}(b) at 1 and I^{∧} denotes the completion of 𝔸^{⟨2ρ,ν⟩} at 0. Furthermore, C_{\underline{𝔾},N_{ν}} is geometrically irreducible and equal to the image of \{1\}ˆ×_{k}I^{∧} in N_{ν}.

In other words, the Newton stratum is up to a finite surjective morphism a product of a closed affine Deligne-Lusztig variety and a central leaf which is an affine space in this case. The dimension of each irreducible component of the Newton stratum of the universal deformation of a local G-shtuka bounded by some μ can be bounded below using the purity of the Newton stratification [HV, Theorem 7.4]. Using this and the fact that the dimension of the central leaves is also known we obtain lower bounds on the dimension of each irreducible component of the affine Deligne-Lusztig varieties. For the affine Deligne-Lusztig varieties in the affine Grassmannian discussed above this implies that they are equidimensional of some dimension which is known by [GHKR1],[Vi1].

Corollary 6.8(a). Let b ∈ LG(k) and let ν be the Newton point of b. Let μ ∈ X_{∗}(T) be dominant with X_{μ}(b) ≠ ∅. Then X_{⪯μ}(b) and X_{μ}(b) are equidimensional of dimension ⟨ρ, μ − ν⟩ − \frac{1}{2}(rk_{𝔽_{q}((z))}(G) − rk_{𝔽_{q}((z))}(J_{b})). In particular, X_{μ}(b) is open and dense in X_{⪯μ}(b).

In [HV] we proved the corollary under the additional assumption that b is basic. Equidimensionality of X_{μ}(b) for b ∈ T (𝔽_{q}((z))) (not necessarily basic) is shown in [GHKR1, Proposition 2.17.1]. For affine Deligne-Lusztig varieties in the affine flag variety the bounds one obtains in this way are in general strictly smaller than the dimension of the variety. In this case there are examples where the affine Deligne-Lusztig varieties are indeed not equidimensional, see [GH]. The results about affine Deligne-Lusztig varieties in the affine flag variety prove a conjecture of Beazley [Be, Conjecture 1].

Acknowledgments. The first author acknowledges support of the DFG (German Research Foundation) in form of DFG-grant HA3006/2-1 and SFB 478, project C5. This work has also been partially supported by SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” of the DFG.

2 Quotients of loop groups and lifting properties

Let O_{S}[[z]] be the sheaf of O_{S}-algebras on S for the fpqc-topology whose ring of sections on an S-scheme Y is the ring of power series O_{S}[[z]](Y) := Γ(Y, O_{Y})[[z]]. This is indeed a sheaf being the countable direct product of O_{S}. Let O_{S}((z)) be the fpqc-sheaf of O_{S}-algebras on S associated with the presheaf Y ↦ Γ(Y, O_{Y})[[z]][\frac{1}{z}]. If Y is quasi-compact then O_{S}((z))(Y) = Γ(Y, O_{Y})[[z]][\frac{1}{z}] by [Hs, Exercise II.1.11].

We consider the ordering ⪯ on the group of coweights X_{∗}(T) of G, which is defined as μ_{1} ⪯ μ_{2} if and only if the difference μ_{2} − μ_{1} is a non-negative integral linear combination of simple coroots. If μ_{1}, μ_{2} are dominant it coincides with the Bruhat ordering. On X^{∗}(T) we consider the analogous ordering. On X_{∗}(T)_{ℚ} = X_{∗}(T) ⊗_{ℤ} ℚ we use the ordering with μ_{1} ⪯ μ_{2} if and only if the difference μ_{2} − μ_{1} is a non-negative rational linear combination of simple coroots. For μ ∈ X_{∗}(T) we denote by z^{μ} the image of z ∈ 𝔾_{m}(𝔽_{q}((z))) under the morphism μ : 𝔾_{m} → T.

For n ≥ 0 let K_{n}, I_{n} ⊂ K_{0} be the subgroup schemes defined by

K_{n}(S) = { g ∈ K_{0}(S) : g ≡ 1 (mod z^{n})}
I_{n}(S) = { g ∈ K_{n}(S) : (g mod z^{n+1}) ∈ B(Γ(S, O_{S})[[z]]/(z^{n+1}))}.

We write I for the Iwahori subgroup scheme I_{0}. We use the letter K for arbitrary subgroup schemes of K_{0} containing some K_{n}, not only for K_{0} itself.

Let LG/K be the sheaf of sets associated with the presheaf S ↦ LG(S)/K(S) on the fppf -site of 𝔽_{q}-schemes S. In particular Gr := LG/K_{0} is called the affine Grassmannian and Flag := LG/I is called the affine flag variety; see [BD, §4.5], [BL], [LS], who comprehensively develop the theory of the affine Grassmannian over the field of complex numbers. Most of their results, and in particular all we use here, also hold with the same proofs over 𝔽_{q}. For instance [NP], [Fa] reprove some statements. Moreover [PR] present a generalization of results and proofs to twisted affine flag varieties over 𝔽_{q}. The sheaf LG/K is represented by an ind-algebraic space over 𝔽_{q} of ind-finite type. Gr and Flag are even ind-schemes. The latter is proved in the literature we just cited by embedding LG/K as a closed ind-scheme of the corresponding ind-scheme for G = GL_{r}. For K = K_{n} and K = I_{n} the same can be done by imposing level structures. In the general case one chooses n such that K_{n} ⊂ K. Then LG/K is the quotient of LG/K_{n} by the smooth group scheme of finite presentation K/K_{n}. Using [LM, Exemple 4.6.1 and Corollaire 8.1.1] one concludes that LG/K is an ind-algebraic space.

We consider right K-torsors and LG-torsors for the étale topology on S; see [HV, §2] for a thorough discussion of this notion.

Lemma 2.1. Let G be a pro-unipotent group and H a connected subgroup such that G/H is finite-dimensional. Then the quotient map G → G/H has a section.

Proof. Note that we do not require H to be normal. As G is pro-unipotent we can choose a finite sequence of subgroups G_{0} = G ⊋ G_{1} ⊋ ⋯ ⊋ G_{n} = H with G_{i}/G_{i+1} ≅ 𝔾_{a} for all i. We see that G/G_{i} → G/G_{i−1} is a 𝔾_{a} ≅ 𝔸^{1}-bundle over G/G_{i−1}. Using induction and that line bundles over an affine space are trivial we obtain G/G_{i} ≅ 𝔸^{i}. As H is pro-unipotent we can choose a sequence of subgroups H_{0} = H ⊋ H_{1} ⊋ ⋯ with ⋂_{i≥0} H_{i} = (1) and H_{i}/H_{i+1} ≅ 𝔾_{a}. Using again induction and also a limit argument we see that it is enough to show that we have a section G/H → G/H_{1}. This follows again from the fact that line bundles over 𝔸^{n} are trivial.

Corollary 2.2. (a) For any n > 0, the projection K_{0} → K_{0}/K_{n} has a section.

(b) Let I′ be a connected subgroup of I containing some K_{n}. Then I → I/I′ has a section.

Proof. For n = 1 this is given by the inclusion K_{0}/K_{1}(S) ≅ G(S) → G(Γ(S, O_{S})[[z]]) ≅ K_{0}(S). Using the short exact sequence

0 → K_{1}/K_{n} → K_{0}/K_{n} → K_{0}/K_{1} → 0

and the section for n = 1 we see that it is enough to prove existence of a section for K_{1} → K_{1}/K_{n}. That follows from the previous lemma. The second part is shown similarly.

Lemma 2.3. Let K ⊂ K_{0} be a subgroup scheme containing some K_{n}. Let S be an 𝔽_{q}-scheme and let h : S → LG/K be a morphism. Then over an étale covering S̃ → S the morphism h lifts to an element h̃ ∈ LG(S̃).

Proof. Let n ∈ ℕ with K_{n} ⊂ K. Then LG/K_{n} is a (K_{0}/K_{n}) ×_{𝔽_{q}} Gr-torsor over the affine Grassmannian Gr and a (K/K_{n}) ×_{𝔽_{q}} (LG/K)-torsor over LG/K. In particular, the projection LG/K_{n} → LG/K is smooth and the map h lifts over an étale covering S′ → S to a morphism h′ : S′ → LG/K_{n}.

\xymatrix{S̃ \ar[r] \ar[d]^{h̃} & S′ \ar[r] \ar[d]^{h′} & S \ar[d]^{h} \ar[dr] \\ LG \ar[r] & LG/K_{n} \ar[r] & LG/K \ar[r] & Gr}

Consider the composite morphism S′ → LG/K_{n} → Gr = LG/K_{0}. By [BD, Theorem 4.5.1(iii)] this morphism lifts to an element h̃ ∈ LG(S̃) over another étale covering S̃ → S′. (Note that in [BD] this is proved to hold even Zariski locally over an algebraically closed field of characteristic zero. Using [NP, Lemma 2.1] the proof carries over for our base field 𝔽_{q} after allowing a finite separable extension of 𝔽_{q}.) The two elements h′ ; S̃ and h̃ of (LG/K_{n})(S̃) have the same image in Gr. Since

(LG/K_{n}) ×_{𝔽_{q}} (K_{0}/K_{n}) ^{∼} ⟶ (LG/K_{n}) ×_{Gr} (LG/K_{n}) , (h, g) ↦ (h, hg)

is an isomorphism, there is an element g̃ ∈ (K_{0}/K_{n})(S̃) with h̃g̃ = h′ ; S̃ in (LG/K_{n})(S̃). Corollary 2.2(a) implies that g̃ lifts to an element of K_{0}(S̃). Replacing h̃ by h̃g̃ finishes the proof.

Let W̃ = N_{T} (k((z)))/T (k[[z]]) be the extended affine Weyl group of G. Here N_{T} is the normalizer of T in G. We have W̃ = X_{∗}(T)×∣W, where W is the finite Weyl group of G. Another decomposition of W̃ is W̃_{0}×∣Ω where Ω ≅ π_{1}(G) consists of the elements x ∈ W̃ with xIx^{−1} = I and where W̃_{0} is a Coxeter group, generated by the (finite and infinite) simple reflections s_{i} associated with our choice of I. A reference for properties of W̃ that are less well known is [HR]. By the affine Bruhat decomposition we have LG(k) = ∐_{x∈W̃} I(k)xI(k). The closure relations between the double cosets are given by the Bruhat ordering on W̃. I.e. the closure of a double coset I(k)xI(k) is the union of the double cosets for y ≤ x. Here y ≤ x if and only if there is a reduced expression x = s_{i_{1}} ⋯ s_{i_{n}}τ with τ ∈ Ω such that y = s_{j_{1}} ⋯ s_{j_{m}}τ for the same τ and such that (j_{1}, …, j_{m}) is a subsequence of (i_{1}, …, i_{n}). There is a length function on W̃ defined by ℓ(x) = n if s_{i_{1}} ⋯ s_{i_{n}}τ is a reduced expression for x. It satisfies ℓ(xy) ≤ ℓ(x) + ℓ(y). The dimension of IxI/I ⊂ Flag is equal to ℓ(x).

Lemma 2.4. For x ∈ W̃ let IxI denote the locally closed subscheme of LG whose k-valued points are I(k)xI(k). Let S be a scheme. Let g ∈ (IxI)(S). Then étale locally on S, there are elements i_{1}, i_{2} of I(S) with g = i_{1}xi_{2}. That is, IxI is the étale sheaf associated with the presheaf S ↦ I(S)xI(S).

Proof. We consider the morphism of ind-schemes I/(I ∩ xIx^{−1}) → Flag given by g ↦ gx. The image consists of the (open) Schubert cell S_{x} = IxI/I. The morphism I/(I ∩ xIx^{−1}) → S_{x} is a bijection on closed points and on tangent spaces, and both schemes are homogenous spaces and hence smooth. Thus I/(I ∩ xIx^{−1}) → Flag is an immersion with image S_{x}. Let g ∈ IxI(S) and let \overline{g} be its image in Flag. Then by what we just saw, \overline{g} is the image of some i ∈ (I/(I ∩ xIx^{−1}))(S). By Corollary 2.2(b) we can lift i to an element of I. This element satisfies x^{−1}i^{−1}g ∈ I(S), the lemma follows.

3 Local G-shtukas

In this section we review and slightly generalize the notions of (bounded) local G-shtukas with level structure and their Newton points from [HV]. Let S be a connected 𝔽_{q}-scheme and let μ be a dominant coweight of G.

Definition 3.1. Let K ⊂ K_{0} be a subgroup scheme containing some K_{n}. Then a local Gshtuka with K-structure over an 𝔽_{q}-scheme S is a pair \underline{G} = (G, φ) where G is a K-torsor over S for the étale topology and φ : σ^{∗}LG ^{∼} ⟶ LG is an isomorphism of LG-torsors over S, where LG is the LG-torsor associated with G.

Later on we are mainly interested in the two cases K = K_{0} and K = I. For P a parabolic subgroup of G we also consider local P -shtukas with K ∩ P -structure. They are defined analogously as K ∩ P -torsors for the étale topology together with a Frobenius map φ on the associated LP -torsor.

Definition 3.2. A quasi-isogeny between local G-shtukas with K-structure (G, φ) → (G′, φ′) over S is an isomorphism of the associated LG-torsors f : LG ^{∼} ⟶ LG′ with φ′σ^{∗}(f) = fφ. The set of quasi-isogenies between \underline{G} and \underline{G}′ over S is denoted QIsog_{S}(\underline{G}, \underline{G}′).

Let k be an algebraically closed field and b ∈ LG(k). Then the quasi-isogeny group J_{b}(𝔽_{q}((z))) of the local G-shtuka with K-structure (K_{k}, bσ^{∗}) can be described analogously to the situation for p-divisible groups as follows.

Proposition 3.3 (Compare [RZ, Propositions 1.12 and 1.16]). Let b_{1}, b_{2} ∈ LG(k). Then the functor on the category of 𝔽_{q}((z))-algebras

Q(R) := { g ∈ G(R ⊗_{𝔽_{q}((z))} k((z))) : gb_{1} = b_{2}σ^{∗}(g) }

is representable by a smooth affine scheme over 𝔽_{q}((z)). For b = b_{1} = b_{2} it is even a group scheme J_{b} := Q.

Assume that b_{1}, b_{2} ∈ LG(k′) for an algebraically closed subfield k′ ⊂ k and let Q′ be the corresponding functor for k′. Then the canonical morphism Q′ → Q is an isomorphism.

In particular if \underline{G}_{i} = (K_{k}, b_{i}σ^{∗}) then QIsog(\underline{G}_{1}, \underline{G}_{2}) = Q(𝔽_{q}((z))). Likewise the quasi-isogeny group QIsog(K_{k}, bσ^{∗}) equals J_{b}(𝔽_{q}((z))). If b_{1}, b_{2}, b ∈ LG(k′) then all these quasi-isogenies are defined over k′.

Corollary 3.4 (Compare [RZ, Corollary 1.14]). Assume that b ∈ LG(k) satisfies a decency equation of the form

(b · σ^{∗})^{s} := b · σ^{∗}(b) · . . . · (σ^{s−1})^{∗}(b) · (σ^{s})^{∗} = z^{γ} · (σ^{s})^{∗}

where s > 0 is an integer and γ ∈ X_{∗}(T), hence satisfies \frac{1}{s}γ_{dom} = ν in X_{∗}(T)_{ℚ} where ν is the Newton point of b. Then J_{b} is an inner form of the centralizer of ν in G_{𝔽_{q^{s}}((z))}, a Levi subgroup of G_{𝔽_{q^{s}}((z))}. In particular QIsog(K_{k}, bσ^{∗}) ⊂ G(𝔽_{q^{s}}((z))) = LG(𝔽_{q^{s}}).

Remark. In the sequel we will shorten (and abuse) notation and simply write J_{b} for the quasi-isogeny group J_{b}(𝔽_{q}((z))) of (K_{k}, bσ^{∗}).

Let \overline{B} ⊂ G be the Borel subgroup opposite to B. For a dominant weight λ of G let V_{λ} := (IndG ; \overline{B}(−λ)_{dom})^{∨} be the Weyl module of G with highest weight λ. It is a cyclic Gmodule generated by a B-stable line on which B acts through λ. Any other such G-module is a quotient of V_{λ}, see for example [Ja, II.2.13]. For a K-torsor G on a scheme S we denote by G_{λ} the fpqc-sheaf of O_{S}[[z]]-modules on S associated with the presheaf

Y ⟼ (G_{K_{0}}(Y) × (V (λ) ⊗_{𝔽_{q}} O_{S}[[z]](Y)))/K_{0}(Y) ;

compare [HV, Section 3]. Here G_{K_{0}} is the K_{0}-torsor associated with G.

Definition 3.5. (a) Let G and H be K-torsors on S and let δ : LH ^{∼} ⟶ LG be an isomorphism of the associated LG-torsors. The isomorphism δ is bounded by μ if for each dominant weight λ of G

δ(H_{λ}) ⊂ z ^{−⟨ (−λ)_{dom},μ⟩} G_{λ} ⊂ G_{λ} ⊗_{O_{S}[[z]]} O_{S}((z)) and
\overline{μ} = \overline{μ}_{δ}(s) in π_{1}(G) for all s ∈ S.

Here μ_{δ}(s) is the unique dominant element with g ∈ K_{0}(k(s))z^{μ_{δ}(s)}K_{0}(k(s)) for some g ∈ LG(k(s)) with δ(H) = g(G) ⊂ LG.

(b) A local G-shtuka with K-structure (G, φ) over S is bounded by μ if the isomorphism

φ : σ^{∗}LG ^{∼} ⟶ LG

is bounded by μ.

(c) Let x ∈ W̃. A local G-shtuka with I-structure \underline{G} over S is of affine Weyl type x if étalelocally on S, it is isomorphic to a trivial I-torsor I_{S} with φ = bσ^{∗} for some b ∈ I(S)xI(S).

Let \underline{G} be a local G-shtuka with K-structure over an algebraically closed field k. Then G_{K_{0}} is isomorphic to the trivial K_{0}-torsor on k and the Frobenius is given by some element b ∈ LG(k). By [HV, Lemma 3.12] there exists some coweight μ such that \underline{G} is bounded by μ. The least such coweight μ\underline{G} is the Hodge polygon or Hodge point of \underline{G}. It is the unique dominant coweight with b ∈ K_{0}(k)z^{μ}K_{0}(k). In [HV, Theorem 5.6, Proposition 5.9, Theorem 9.5, Remark 9.4 (b), and Proposition 3.16] (for the case where ζ = 0) we proved the following proposition for K = K_{0}, respectively K = I. Its generalization to arbitrary K is proved in the same way.

Proposition 3.6. Let \underline{𝔾} = (K_{k′}, bσ^{∗}) be a local G-shtuka with K-structure over a field k′ bounded by μ (respectively of affine Weyl type y if K = I). Then the universal deformation of \underline{𝔾} by local G-shtukas with K-structure bounded by μ (respectively of affine Weyl type y) is pro-representable by a complete noetherian local ring \overline{D} of dimension ⟨2ρ, μ⟩ (respectively of dimension ℓ(y), the length of y). The universal local G-shtuka with K-structure over Spf \overline{D} comes from a local G-shtuka with K-structure over Spec \overline{D}.

Finally we will need the following fact which is a straightforward generalization of [HV, Theorem 6.2] (in the case where ζ = 0).

Proposition 3.7. Let X be an 𝔽_{q}-scheme. Let b ∈ LG(X) and consider the local G-shtuka with K-structure H = (K_{X}, bσ^{∗}) over X. Then the ind-scheme (LG/K) ×_{𝔽_{q}} X pro-represents the functor from X-schemes to sets

S ⟼ \{ Isomorphism classes of pairs (\underline{G}, ρ) where \underline{G} is a local G-shtuka
with K-structure over S and ρ : \underline{G} → \underline{H}_{S} is a quasi-isogeny \}

Here (\underline{G}, ρ) and (\underline{G}′, ρ′) are called isomorphic if there is an isomorphism α : \underline{G} ^{∼} ⟶ \underline{G}′ with ρ = ρ′ ∘ α.

4 Completely slope divisible local G-shtukas

For a parabolic subgroup P ⊂ G containing T but not necessarily B let M be the Levi factor containing T and let N be the unipotent radical of P. Let \overline{P} be the opposite parabolic and \overline{N} its unipotent radical. Denote by W̃_{M} the extended affine Weyl group of M. It is canonically a subgroup of W̃.

Let K be a subgroup scheme of I. We define the group schemes K_{M}, K_{N}, and K\overline{N} by their values on 𝔽_{q}-algebras R

K_{M}(R) := K(R) ∩ M(R[[z]]) , K_{N}(R) := I(R) ∩ N(R[[z]]) , K\overline{N}(R) := I(R) ∩ \overline{N}(R[[z]]) .

Let us recall the definition and some of the main properties of fundamental alcoves from [GHKR2].

Definition 4.1 ([GHKR2, Definition 13.1.2]). An element x ∈ W̃ is called P -fundamental, or a fundamental P -alcove if x ∈ W̃_{M} and if xI_{M}x^{−1} = I_{M}, xI_{N}x^{−1} ⊂ I_{N}, and x^{−1}I\overline{N}x ⊂ I\overline{N}.

Remark 4.2. (a) [GHKR2, Corollary 13.2.4] Let b ∈ LG(k) for some algebraically closed field k. Then there is an x ∈ W̃ which is P -fundamental for some P and such that b = g^{−1}xσ^{∗}(g) for some g ∈ LG(k).

(b) [GHKR2, Proposition 6.3.1] If x is P -fundamental, and g ∈ I(k)xI(k) then there is an h ∈ I(k) with h^{−1}gσ^{∗}(h) = x.

(c) [GHKR2, Corollary 13.2.4], [Vi4, Lemma 9] All elements x as in (a) are obtained as follows. Let ν be the Newton point of b and let M_{ν} be the centralizer of ν. Then the σ-conjugacy class [b] of b contains a uniquely determined element x_{b} ∈ W̃_{M_{ν}} such that x_{b} has length zero with respect to M_{ν} and I ∩M_{ν}, and the M_{ν}-dominant Newton point of x_{b} equals ν. It is called the standard representative of [b]. Let P be the standard parabolic subgroup with Levi component M_{ν}. Then there is an element w ∈ W of minimal length in its coset W_{M_{ν}} · w such that x := w^{−1}x_{b}w ∈ [b] is w^{−1}P w-fundamental, and all x ∈ [b] that are P -fundamental for some P arise in this way.

(d) [Vi4, Lemma 9 (c)] Let x be P -fundamental, and M′ the centralizer of its M-dominant Newton point. Then M′ ⊇ M and x is also P′-fundamental for P′ = M′P.

(e) Let x ∈ W̃_{M} be P -fundamental for P = MN and assume that M is the centralizer of the M-dominant Newton point ν_{x} of x. Then for any d ∈ ℕ there is an l ∈ ℕ with

x^{l}I_{N}x^{−l} , x^{−l}I\overline{N}x^{l} ⊂ I_{d}.

Indeed, let t > 0 such that x^{t} ∈ X_{∗}(T) ⊂ W̃_{M}. Then by our choice of M, we have ⟨α, x^{t}⟩ = ⟨α, tν⟩ > 0 for all roots α of N and ⟨ᾱ, x^{t}⟩ < 0 for all roots ᾱ of \overline{N}.

Lemma 4.3. Let x be a P -fundamental alcove where P is such that the Levi subgroup M of P equals the centralizer of the M-dominant Newton point of x. Let \overline{N} be the unipotent radical of the parabolic opposite to P and let n̄ ∈ L\overline{N}(S) for a quasi-compact scheme S. Then there exists an integer l_{0} ∈ ℤ with n̄ ∈ x^{−l_{0}}I\overline{N}(S)x^{l_{0}}. In particular n̄ is bounded.

Proof. Since S is quasi-compact L\overline{N}(S) = \overline{N}(Γ(S, O_{S}((z)))) = \overline{N}(Γ(S, O_{S})[[z]][\frac{1}{z}]). Again by the quasi-compactness of S, the argument given in Remark 4.2(e) implies that x^{tl}n̄x^{−tl} ∈ I\overline{N} for l ≪ 0. Also the boundedness of n̄, that is of the automorphism δ of LG_{S} given by left multiplication by n̄, follows from this and from the fact that μ̄_{δ}(s) = 0 for all s ∈ S.

Example 4.4. Let G = GL_{5} with B being the Borel of upper triangular matrices, and let b = (12)(354)z^{(1,0,1,0,0)}. Then b has Newton point (\frac{1}{2}, \frac{1}{2}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}) and is already the standard representative of its class [b]. In this example, the only fundamental alcove in [b] is x = (13)(254)z^{1,1,0,0,0} = w^{−1}bw for w = (23). In particular, we see that we cannot choose x to be P -fundamental for a standard P.

Lemma 4.5. Let b ∈ LG(k) and let x be a P -fundamental alcove in the σ-conjugacy class [b]. Then IxI ⊂ [b] is closed in [b].

Proof. Recall that the closure of IxI is the union of all IyI with y ≤ x in the Bruhat order. Assume that there is some g ∈ IyI with y ≤ x and g ∈ [b]. Then by [GHKR2, Corollary 12.1.2] or [Vi4, Proposition 12], there is a w ∈ W with w^{−1}xw ≤ y. By [GHKR2, Proposition 13.1.2] we have ℓ(x^{n}) = n⟨2ρ, ν_{b}⟩ = nℓ(x) for each n ∈ ℕ where ρ is the half-sum of the positive roots and where ν_{b} is the Newton point of b. Furthermore nℓ(w^{−1}xw) ≥ ℓ((w^{−1}xw)^{n}) = ℓ(w^{−1}x^{n}w) ≥ nℓ(x) − 2ℓ(w) for each n, hence ℓ(w^{−1}xw) ≥ ℓ(x). Thus w^{−1}xw = y = x.

The following lemma is a scheme-theoretic version of a special case of the Iwahori decomposition.

Lemma 4.6. Let K be a subgroup scheme of I such that K(S) ⊃ T (Γ(S, O_{S})[[z]]) for every 𝔽_{q}-scheme S. Then every element g ∈ K(S) has uniquely determined decompositions g = nmn̄ and g = n̄′m′n′ for elements m, m′ ∈ K_{M}(S), n, n′ ∈ K_{N}(S), and n̄, n̄′ ∈ K\overline{N}(S).

Proof. As K(k) ⊃ T (k[[z]]), the group scheme K is generated by T (k[[z]]) and all affine root subgroups in K. There is a filtration K = G_{1} ⊃ G_{2} ⊃ G_{3} ⊃ ⋯ such that G_{2} is the unipotent radical of K, each G_{i} is a normal subgroup of K defined over 𝔽_{q} and for i ≥ 2, we have that G_{i}/G_{i+1} ≅ 𝔾_{a} is generated by one of the affine root subgroups. Indeed, it can be obtained by intersecting a suitable filtration of I by Moy-Prasad subgroups with K. We first decompose g as g_{1}g_{2} with g_{1} ∈ T (Γ(S, O_{S})[[z]]) ⊂ K_{M}(S) and g_{2} ∈ G_{2}(Γ(S, O_{S})[[z]]). As each affine root subgroup is contained in one of the subgroups K_{M}, K_{N}, K\overline{N}, one can then inductively construct decompositions of g modulo some G_{i}. By passing to the limit one obtains the decompositions above. The uniqueness follows from the fact \overline{N} ∩ MN = \{1\}.

Definition 4.7. A bounded local G-shtuka with I-structure \underline{G} over an 𝔽_{q}-scheme S is called completely slope divisible if there exists a parabolic subgroup P of G containing T, a local \overline{P} -shtuka with I ∩ L\overline{P} -structure \underline{P̄} = (P̄, φ) over S, and a P -fundamental x such that

• \underline{G} ≅ i_{∗}\underline{P̄} where i : \overline{P} ↪ G is the inclusion morphism,

• étale-locally on S the Frobenius φ is represented by an element of x · (L\overline{P} ∩ I)(S).

Note that by Remark 4.2 (d) one can always enlarge M, P and \overline{P} and assume that M equals the centralizer of the M-dominant Newton point of x. See Section 5 for a comparison to the corresponding (but slightly weaker) notion for p-divisible groups defined in [OZ, Def 1.2].

Proposition 4.8. Let x be P -fundamental for a parabolic P = MN ⊃ T in which M is the centralizer of the M-dominant Newton point of x. Let g ∈ I(S)xI(S) for an 𝔽_{q}-scheme S. Then there is an element h ∈ I(S) with h^{−1}gσ^{∗}(h) = xp for p ∈ I_{M}(S)I\overline{N}(S). In other words, the local G-shtuka with I-structure defined by g is completely slope divisible.

Proof. By induction on l we prove the following Claim. For any l ∈ ℕ there exists an element h_{l} ∈ I(S) with h_{l}^{−1}gσ^{∗}(h_{l}) = xp_{l}r_{l} for elements p_{l} ∈ I_{M}(S)I\overline{N}(S) and r_{l} ∈ x^{l}I_{N}(S)x^{−l}. We can choose the h_{l} such that h_{l−1}^{−1}h_{l} ∈ x^{l}I_{N}(S)x^{−l}.

To prove the claim for l = 0 we write g = h_{0}xh′_{0} with h_{0}, h′_{0} ∈ I(S). Then h_{0}^{−1}gσ^{∗}(h_{0}) = xh′_{0}σ^{∗}(h_{0}) with h′_{0}σ^{∗}(h_{0}) ∈ I(S) = I_{M}(S)I\overline{N}(S)I_{N}(S) by Lemma 4.6. Now assume that the claim holds for l. By Lemma 4.6 for the group K = I ∩ x^{l}Ix^{−l} = (x^{l}I_{N}x^{−l})I_{M}I\overline{N} we write p_{l}r_{l} = n_{l}m_{l}n̄_{l} with n_{l} ∈ x^{l}I_{N}(S)x^{−l}, m_{l} ∈ I_{M}(S), and n̄_{l} ∈ I\overline{N}(S). Setting h_{l+1} := h_{l}xn_{l}x^{−1} we compute

h_{l+1}^{−1}gσ^{∗}(h_{l+1}) = xn_{l}^{−1}x^{−1}h_{l}^{−1}gσ^{∗}(h_{l})xσ^{∗}(n_{l})x^{−1} = xm_{l}n̄_{l}xσ^{∗}(n_{l})x^{−1} = xp_{l+1}r_{l+1}

for p_{l+1} = m_{l}n̄_{l} ∈ I_{M}(S)I\overline{N}(S) and r_{l+1} = xσ^{∗}(n_{l})x^{−1} ∈ x^{l+1}I_{N}(S)x^{−(l+1)}. This proves the claim.

By Remark 4.2 (e), we have that for every d ∈ ℕ there is an l ∈ ℕ such that for all integers l′ ≥ l

h_{l}^{−1}h_{l′} ∈ x^{l}I_{N}(S)x^{−l} ⊂ K_{d}.

This implies that the sequence h_{l} converges to an element h ∈ I(S) in the z-adic topology. Also M\overline{N} ∩ N = \{1\} implies that r_{l′} ≡ r_{l} ≡ 1 (mod z^{d}) and p_{l′} ≡ p_{l}. Hence the p_{l} converge to an element p ∈ I_{M}(S)I\overline{N}(S) and the r_{l} converge to 1. Altogether we find h^{−1}gσ^{∗}(h) = xp as desired.

The proposition implies the following characterization of complete slope divisibility (use Remark 4.2 (e)).

Corollary 4.9. A bounded local G-shtuka with I-structure \underline{G} over an 𝔽_{q}-scheme S is completely slope divisible if and only if there is a P -fundamental element x ∈ W̃ for some parabolic P ⊃ T such that étale-locally on S, \underline{G} is isomorphic to (I_{S}, gσ^{∗}) with g ∈ I(S)xI(S).

Remark 4.10. Let \underline{G} be a completely slope divisible local G-shtuka with I-structure over an algebraically closed field k and let x be the associated fundamental alcove. Then by Remark 4.2 (b), there is a trivialization of the I-torsor such that the Frobenius is given by xσ^{∗}.

Lemma 4.11. Let x ∈ W̃_{M} be P -fundamental for P = MN where M is the centralizer of the M-dominant Newton point of x. Let n̄ ∈ L\overline{N}(S) for a noetherian 𝔽_{q}-scheme S and fix some d ∈ ℕ. Then n̄ is bounded, i.e. contained in x^{−l_{0}}I\overline{N}(S)x^{l_{0}} for some l_{0} ∈ ℤ, and there is a finite surjective radicial morphism S′ → S and an element h ∈ x^{−l_{0}}I\overline{N}(S′)x^{l_{0}} with h^{−1}xn̄σ^{∗}(h) = xn̄′ for n̄′ ∈ I\overline{N}(S′) ∩ I_{d}(S′). If S is local with closed point s and if the reduction of n̄ in s is equal to 1, then h can be chosen such that its reduction in s is also equal to 1.

Proof. The boundedness of n̄ and the existence of l_{0} ∈ ℤ with n̄ ∈ x^{−l_{0}}I\overline{N}(S)x^{l_{0}} follows from Lemma 4.3. We use induction on l ≥ l_{0} and construct finite coverings S_{l} → S and elements h_{l} ∈ x^{−l_{0}}I\overline{N}(S_{l})x^{l_{0}} with h_{l}^{−1}xn̄σ^{∗}(h_{l}) = xn̄_{l} such that n̄_{l} = x^{−l}y_{l}x^{l} ∈ x^{−l}I\overline{N}(S_{l})x^{l}. Choosing l big enough suffices for our purpose by Remark 4.2 (e). We start with S_{l_{0}} = S, h_{l_{0}} = 1, and n̄_{l_{0}} = n̄. Assume we already found S_{l} and h_{l}. Since I\overline{N}/x^{−1}I\overline{N}x is a scheme of finite type over 𝔽_{q}, its q-Frobenius endomorphism σ is finite surjective. Consider the cartesian diagram in which the lower horizontal morphism is given by the element y_{l}^{−1} ∈ I\overline{N}(S_{l})

[figure: S_{l+1}; ỹ_{l}^{−1}; I\overline{N}/x^{−1}I\overline{N}x; σ; S_{l}; y_{l}^{−1}; I\overline{N}/x^{−1}I\overline{N}x]

Then S_{l+1} → S_{l} is also a finite surjective radicial morphism. The upper horizontal morphism is given by an element ỹ_{l}^{−1} ∈ I\overline{N}(S_{l+1}) with σ^{∗}(ỹ_{l}^{−1}) = y_{l}^{−1}(x^{−1}f_{l}x) for an f_{l} ∈ I\overline{N}(S_{l}). Set h_{l+1} = h_{l}x^{−l}ỹ_{l}^{−1}x^{l}. Then

h_{l+1}^{−1}xn̄σ^{∗}(h_{l+1}) = x^{−l}ỹ_{l}x^{l}xx^{−l}y_{l}x^{l}x^{−l}σ^{∗}(ỹ_{l}^{−1})x^{l}
= x(x^{−(l+1)}ỹ_{l}f_{l}x^{l+1}).

Let y_{l+1} := ỹ_{l}f_{l} ∈ I\overline{N}(S_{l+1}). Then

n̄_{l+1} = x^{−(l+1)}y_{l+1}x^{l+1} ∈ x^{−(l+1)}I\overline{N}(S_{l+1})x^{l+1}

as desired. Let l be big enough such that x^{−l}I\overline{N}x^{l} ⊂ I_{d}. We stop the iteration at l to find a finite surjective radicial morphism S′ := S_{l} → S, an element h := h_{l} ∈ x^{−l_{0}}I\overline{N}(S′)x^{l_{0}} with h^{−1}xn̄σ^{∗}(h) = xn̄′ for n̄′ := n̄_{l} ∈ I\overline{N}(S′) ∩ I_{d}(S′). The last assertion also follows from the explicit construction of the elements h_{l}.

Over a perfect field k the proof has the following

Corollary 4.12. Let x ∈ W̃_{M} be P -fundamental for P = MN where M is the centralizer of the M-dominant Newton point of x. Let n̄ ∈ x^{−l_{0}}I\overline{N}(k)x^{l_{0}} for a perfect field k and some l_{0} ∈ ℤ. Then there is an element h ∈ x^{−l_{0}}I\overline{N}(k)x^{l_{0}} with h^{−1}xn̄σ^{∗}(h) = x.

Proof. In the proof of Lemma 4.11 we have S_{l} = Spec k for all l with the morphisms S_{l+1} → S_{l} being the Frobenius automorphism. Here we do not stop the iteration. Instead we let h be the limit of the h_{l}. It exists because the x^{−l}ỹ_{l}^{−1}x^{l} ∈ x^{−l}I\overline{N}(k)x^{l} and the \overline{n}_{l} ∈ x^{−l}I\overline{N}(k)x^{l} converge to 1.

Proposition 4.13. Let \underline{H} be a local G-shtuka with K-structure over an algebraically closed field k. Let μ ∈ X_{∗}(T) be dominant. Then up to isomorphism there exist only finitely many quasi-isogenies ρ : \underline{G} → \underline{H} bounded by μ with \underline{G} a completely slope divisible local G-shtuka with K-structure. Here we say that ρ : \underline{G} → \underline{H} and ρ′ : \underline{G}′ → \underline{H} are isomorphic if there is an isomorphism α : \underline{G} ^{∼} ⟶ \underline{G}′ with ρ = ρ′ ∘ α.

Proof. By Remark 4.2 (a) there is a quasi-isogeny η : \underline{H} → \underline{H̃} with \underline{H̃} = (K_{k}, xσ^{∗}) for some P -fundamental x. By [HV, Lemma 3.11] the quasi-isogeny η is bounded by its Hodge point μ_{η}. If ρ : \underline{G} → \underline{H} is a quasi-isogeny bounded by μ then η ∘ ρ is bounded by μ + μ_{η}. Hence we may assume that \underline{H} = \underline{H̃}. Note that there are only finitely many possibilities for the parabolic subgroup P. By Corollary 4.9 and Remark 4.2 (b) and (d) we may assume that \underline{G} = (K_{k}, x′σ^{∗}) for some P′-fundamental x′ such that P′ is the centralizer of the M′-dominant Newton point of x′. As \underline{G} and \underline{H} are isogenous, Remark 4.2 (c) and (d) imply that we may assume that x = x′ and P = P′. Then the quasi-isogeny is given by an element h ∈ QIsog(K_{k}, xσ^{∗}) which is bounded by μ. Since x is decent, Corollary 3.4 shows that there are only finitely many such quasi-isogenies.

Theorem 4.14. Let S be an integral noetherian 𝔽_{q}-scheme and let g ∈ LG(S) such that the σ-conjugacy classes of g_{s} in the geometric points s of S all coincide. Let x be a P -fundamental alcove associated with this σ-conjugacy class as in Remark 4.2 (a). Then there are morphisms S̃ ⟶^{β} S′ ⟶^{α} S with S′ integral, α finite surjective, and β an étale covering, and a bounded element h̃ ∈ LG(S̃) satisfying h̃^{−1}gS̃σ^{∗}(h̃) ∈ I(S̃)xI(S̃).

The Theorem is an analog of [OZ, Lemma 2.4].

Proof. Let \overline{L} be an algebraic closure of the function field L of S. By Remark 4.2 (a) there is an h̄ ∈ LG(\overline{L}) with h̄^{−1}g\overline{L} σ^{∗}(h̄) = x. The image of h̄ in the affine flag variety lies inside a closed Schubert cell \overline{C} ⊂ Flag. Note that \overline{C} is a scheme which is projective over 𝔽_{q}. Hence by multiplying h̄ with an element of I(\overline{L}) on the right we may assume that there is a finite field extension L′/L and an h′_{L′} ∈ LG(L′) with (h′_{L′})^{−1}g_{L′} σ^{∗}(h′_{L′}) ∈ I(L′)xI(L′). Let S′ be the scheme theoretic closure of the L′-valued point h′_{L′} inside \overline{C} ×_{𝔽_{q}} S, and let α : S′ → S and h′ : S′ → \overline{C} ⊂ Flag be the projections. Then S′ is irreducible and reduced. Over an étale covering β : S̃ → S′ the map h′ is represented by an element h̃ ∈ LG(S̃) due to Lemma 2.3.

Consider the S̃-valued point h̃^{−1}gS̃ σ^{∗}(h̃) of the affine flag variety Flag. By construction it lies generically in IxI/I. So it lies inside the closed Schubert cell \overline{IxI/I} ⊂ Flag. By assumption it lies in the Newton stratum of x inside \overline{IxI/I}. By Lemma 4.5 IxI/I is closed in this Newton stratum. Hence the point lies in IxI. By Lemma 2.4 we find that h̃^{−1}gS̃ σ^{∗}(h̃) lies in I(S̃)xI(S̃).

It remains to show that the morphism α : S′ → S is finite. For this consider the Stein factorization S′ → S′′ → S with S′′ = Spec α_{∗}O_{S′}. Then S′′ → S is finite and γ : S′ → S′′ has connected fibers [EGA, III, Corollaire 4.3.2]. We reduce to the following

Claim. For any point s ∈ S′′ the fiber S′_{s} = γ^{−1}(s) is mapped under h′ : S′ → \overline{C} to a single point in \overline{C}.

Indeed, by [OZ, Lemma 2.6] the claim implies that the morphism h′ : S′ → \overline{C} factors through S′′, and thus S′ = S′′ by definition of S′. Let us prove the claim. Consider the local Gshtuka with I-structure \underline{H} = (I_{S}, gσ^{∗}) over S. By Proposition 3.7 the map h′ : S′ → Flag uniquely determines a local G-shtuka with I-structure \underline{G} over S′ together with a quasi-isogeny ρ : \underline{G} → \underline{H}_{S′} up to isomorphism. Its pullback to S̃ is trivialized GS̃ ≅ (IS̃, h̃^{−1}gS̃ σ^{∗}(h̃)σ^{∗}) and ρ is given by multiplication with h̃; compare the proof of [HV, Theorem 6.2]. The quasi-isogeny ρ is bounded by some dominant μ ∈ X_{∗}(T) because h′ lies in the Schubert cell \overline{C}(S′). Since h̃^{−1}gS̃ σ^{∗}(h̃) ∈ I(S̃)xI(S̃), Corollary 4.9 tells us that \underline{G} is completely slope divisible. Now fix a point s ∈ S′′ and consider the base changes of \underline{H} to the residue field k(s), and of \underline{G} to the fiber S′_{s} over s. Then we are in the situation of Proposition 4.13 and we conclude that the image of h′ : S′_{s} → \overline{C} is a finite set of points. Since S′_{s} is connected, this image must be a single point in \overline{C}. This proves the claim and the theorem.

Corollary 4.15. Let R be an integral noetherian complete local ring with algebraically closed residue field k. Let \underline{G} be a local G-shtuka over Spec R with K-structure whose quasi-isogeny class in all geometric points of R is constant. Let x be a P -fundamental alcove corresponding to this quasi-isogeny class. Let ρ_{0} be a quasi-isogeny between the fiber of \underline{G} over k and (K_{k}, xσ^{∗}). Then for each fixed d ∈ ℕ there exists a finite integral extension R′′ ⊃ R and a bounded quasiisogeny ρ : \underline{H} → \underline{G}_{R′′} of local G-shtukas with K-structure over Spec R′′ with reduction ρ_{k} = ρ_{0}, where \underline{H} has a trivialization \underline{H} ≅ (K_{R′′}, x\overline{n}σ^{∗}) with \overline{n} ∈ I_{d,\overline{N}}(R′′) and reduction \overline{n}_{k} = 1.

Note that for K = I, the local G-shtuka with I-structure \underline{H} is completely slope divisible, compare Corollary 4.9.

Proof. Since R has no non-trivial étale coverings, there is a trivialization \underline{G} = (K_{R}, gσ^{∗}). Thus the assertion is equivalent to the existence of R′′ and an element h ∈ LG(R′′) with reduction ρ_{0} over the special point such that h^{−1}gσ^{∗}(h) = x\overline{n} with \overline{n} ∈ I_{d,\overline{N}}(R′′). Note that by [HV, Lemma 3.13] the boundedness of h and ρ is automatic because R′′ is integral.

Theorem 4.14 yields a finite surjective morphism S′ = Spec R′ → S = Spec R with R′ integral and an element h_{1} ∈ LG(R′) = G(R′((z))) with b′ := (h_{1})^{−1}g_{R′}σ^{∗}(h_{1}) ∈ I(R′)xI(R′).

Indeed, observe that the étale morphism β : S̃ → S′ from Theorem 4.14 is an isomorphism because also R′ is complete with algebraically closed residue field. By Proposition 4.8 we may further assume that b′ = xmn̄_{1} ∈ x I_{M}(R′)I\overline{N}(R′) where M is the centralizer of the Mdominant Newton point of x and P ⊂ MN. Since xm is basic in M, [HV, Proposition 8.1] yields an element h_{2} ∈ M(R′((z))) which σ-conjugates xm to x. Hence h_{2}^{−1}b′σ^{∗}(h_{2}) = xn̄′ for n̄′ := σ^{∗}(h_{2})^{−1}n̄_{1}σ^{∗}(h_{2}) ∈ \overline{N}(R′((z))). Let n̄′_{k} be the reduction of n̄′ in the special point. Then n̄′_{k} ∈ x^{−l_{0}}I\overline{N}(k)x^{l_{0}} for some l_{0} ∈ ℤ by Lemma 4.3. Corollary 4.12 shows that there is an h_{3,k} ∈ x^{−l_{0}}I\overline{N}(k)x^{l_{0}} with h_{3,k}^{−1}xn̄′_{k}σ^{∗}(h_{3,k}) = x. We lift the element h_{3,k} to an element h_{3} ∈ x^{−l_{0}}I\overline{N}(R′)x^{l_{0}}. This is possible since I\overline{N} is pro-unipotent. Replacing h_{2} by h_{2}h_{3} we see that we may in addition assume that n̄′_{k} = 1. The given quasi-isogeny ρ_{0} corresponds to an element h ∈ LG(k) with j_{0} = h^{−1}(h_{1}h_{2})_{k} ∈ J_{x}(k). There is a lift j ∈ J_{x} ∩LG(R′) of j_{0} because x ∈ LG(𝔽_{q}) is decent, hence j_{0} ∈ LG(𝔽_{q^{s}}) by Corollary 3.4, and 𝔽_{q^{s}} ⊂ R′ since k contains 𝔽_{q}^{alg} and R′ is henselian. Replacing h_{1}h_{2} by h_{1}h_{2}j^{−1} we see that we may in addition assume that (h_{1}h_{2})_{k} yields the given isogeny ρ_{0} between \underline{G}_{k} and (K_{k}, xσ^{∗}). We now apply Lemma 4.11 for n̄′ (with some possibly different l_{0}) and the constant d and obtain a finite extension R′′ of R′ and an element h_{4} ∈ \overline{N}(R′′((z))) with (h_{4})_{k} = 1 such that h_{4}^{−1}xn̄′σ^{∗}(h_{4}) = x\overline{n} with \overline{n} ∈ I_{d,\overline{N}}(R′′). Setting h = h_{1}h_{2}h_{4} proves the assertion.

5 The general linear group

In this section we compare our notion of completely slope divisible local G-shtukas to Zink’s notion of completely slope divisible p-divisible groups. To make the parallel more visible we also explain the relation to local shtukas (which are described via locally free sheaves and correspond to local G-shtukas for G = GL_{r}). We denote by σ^{∗} the endomorphism of O_{S}[[z]] and O_{S}((z)) that acts as the identity on the variable z, and as b ↦ b^{q} on local sections b ∈ O_{S}. For a sheaf V of O_{S}[[z]]-modules on S we set σ^{∗}V := V ⊗_{O_{S}[[z]],σ^{∗}} O_{S}[[z]].

Definition 5.1. A local shtuka over an 𝔽_{q}-scheme S (of rank r) is a pair \underline{V} = (V, φ) where V is a sheaf of O_{S}[[z]]-modules on S which Zariski-locally is free of rank r, together with an isomorphism of O_{S}((z))-modules φ : σ^{∗}V ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ V ⊗_{O_{S}[[z]]} O_{S}((z)).

A quasi-isogeny between local shtukas (V, φ) → (V′, φ′) is an isomorphism of O_{S}((z))- modules f : V ⊗_{O_{S}[[z]]} O_{S}((z)) ^{∼} ⟶ V′ ⊗_{O_{S}[[z]]} O_{S}((z)) with φ′σ^{∗}(f) = fφ.

Lemma 5.2 ([HV, Lemma 4.2]). The quasi-isogeny categories of local GL_{r}-shtukas with K_{0}- structure over S and of local shtukas over S are equivalent. In particular the trivialized local GL_{r}-shtuka with K_{0}-structure (K_{0,S}, gσ^{∗}) over S corresponds to the trivialized local shtuka (O_{S}[[z]]^{⊕r}, φ = gσ^{∗}).

Definition 5.3. For general G a local G-shtuka with K_{0}-structure \underline{G} = (G, φ) over an 𝔽_{q}- scheme S is called completely slope divisible in the sense of Zink if there exists a standard parabolic subgroup P ⊂ G with opposite parabolic \overline{P}, a local \overline{P} -shtuka with L\overline{P} ∩K_{0}-structure \underline{P̄} = (P̄, φ) over S, an integer s > 0, and a central cocharacter μ : 𝔾_{m} → M which is G-dominant (M is the Levi component of P containing T) such that

(a) \underline{G} ≅ i_{∗}\underline{P̄} where i : \overline{P} ↪ G is the inclusion morphism, (b) If \underline{P̄}_{S′} is trivializable for some S′ → S then with respect to every trivialization, z^{−μ}φ^{s} is

represented by an element of \overline{P} (Γ(S′, O_{S′})[[z]]).

Assume that the second condition holds for some trivialization of \underline{G}_{S′}. The element b obtained from any other trivialization differs from the old by σ-conjugation with an element of \overline{P} (Γ(S′, O_{S′})[[z]]). Thus it automatically also satisfies the second condition. Also, being completely slope divisible in Zink’s sense is a local property.

Proposition 5.4. Let G = GL_{r}. Then a local GL_{r}-shtuka with K_{0}-structure \underline{G} over S is completely slope divisible in the sense of Zink if and only if its associated local shtuka \underline{V} = (V, φ) satisfies the following condition: There is a sequence of local quotient-shtukas \underline{V} = \underline{V}_{e} → → \underline{V}_{e−1} → → … → → \underline{V}_{1} → → \underline{V}_{0} = (0) and integers t_{1} > … > t_{e} and s > 0 such that for all 1 ≤ i ≤ e

(a) z^{−t_{i}}φ^{s}((σ^{s})^{∗}V_{i}) ⊂ V_{i} and

(b) on W_{i} := ker(V_{i} → → V_{i−1}) the map z^{−t_{i}}φ^{s} : (σ^{s})^{∗}W_{i} ^{∼} ⟶ W_{i} is an isomorphism.

Note that the latter condition is indeed the same as complete slope divisibility in [OZ, Definition 1.2] when one views local shtukas as analogs of the contravariant Dieudonné modules of p-divisible groups.

Proof. Assume that \underline{G} is completely slope divisible in the sense of Zink and let P, M, μ, s, \underline{P̄} be as in Definition 5.3. We may even assume that M is the centralizer of μ after possibly enlarging M and P. We pass to an étale covering S′ → S in order to choose trivializations of \underline{P̄} and \underline{G} such that z^{−μ}φ^{s} = p̄σ^{∗} with p̄ ∈ \overline{P} (Γ(S′, O_{S′})[[z]]). By our choice of B as the upper triangular matrices, M consists of all block diagonal matrices of fixed block sizes d_{1}, d_{2}, …, d_{e} and \overline{P} consists of the corresponding lower triangular block matrices. The successive projections onto the upper left blocks uniquely defines a sequence of local quotientshtukas \underline{V}_{S′} = \underline{V}_{e} → → \underline{V}_{e−1} → → … → → \underline{V}_{1} → → \underline{V}_{0} = (0) over S′. Namely V_{i} is spanned by the first d_{1} + … + d_{i} basis vectors. By its uniqueness this sequence descends to S. The GL_{r}-dominant cocharacter μ can be written as

μ = (t_{1}, …, t_{1} , t_{2}, …, t_{2} , …, t_{e}, …, t_{e})

with t_{1} > t_{2} > … > t_{e} since M is the centralizer of μ. Then z^{−t_{i}}φ^{s} = z^{μ−t_{i}} p̄ and this implies (a) and (b) because μ is central in M.

Conversely assume that \underline{G} satisfies the condition of the proposition. Over a Zariski-covering S′ → S we can successively choose compatible trivializations of all W_{i} and V_{i} as free O_{S′}[[z]]- modules. We order the obtained basis vectors of V such that the first d_{1} + … + d_{i} of them project onto a basis of V_{i}, where d_{i} = rk W_{i}. With respect to this basis the Frobenius of \underline{V} is given by an element p̄′ ∈ L\overline{P} (S′) for \overline{P} the parabolic consisting of the lower triangular block matrices whose Levi component M contains all block diagonal matrices with block sizes d_{1}, …, d_{e}. The opposite parabolic P is standard. The local \overline{P} -shtuka with L\overline{P} ∩ K_{0}-structure \underline{P̄}′ = ((L\overline{P} ∩ K_{0})_{S′}, p̄′σ^{∗}) over S′ descends to a local \overline{P} -shtuka \underline{P̄} over S which satisfies \underline{G} ≅ i_{∗}\underline{P̄}. Let μ be the GL_{r}-dominant cocharacter with weights t_{i} of multiplicity d_{i}. Then μ : 𝔾_{m} → T ⊂ M is central in M. By (a) the Frobenius z^{−μ}φ^{s} = z^{−μ}p̄′σ^{∗}(p̄′)·. . .·(σ^{s−1})^{∗}(p̄′) lies in Γ(S′, O_{S′})[[z]]^{r×r} By (b) its Levi part, that is its semi-simplification, lies in M(Γ(S′, O_{S′})[[z]]). Altogether this implies that z^{−μ}φ^{s} ∈ \overline{P} (Γ(S′, O_{S′})[[z]]) and the proposition is proved.

Lemma 5.5. A local G-shtuka with K_{0}-structure \underline{G} over S which is completely slope divisible (in the sense of Definition 4.7) is completely slope divisible in the sense of Zink.

Proof. As both conditions are local on S we may replace S by an étale covering and assume that G is trivializable. Since \underline{G} is completely slope divisible we may choose a trivialization \underline{G} ≅ (K_{0,S}, bσ^{∗}) with b = xmn̄ and m ∈ I_{M}(S), n̄ ∈ I\overline{N}(S). Then

φ^{s} = x^{s} · x^{−(s−1)}mn̄x^{s−1} · . . . · x^{−1}(σ^{s−2})^{∗}(mn̄)x · (σ^{s−1})^{∗}(mn̄) =: x^{s} · p̄.

Since x is P -fundamental we obtain p̄ ∈ I_{M}(S)I\overline{N}(S) ⊂ \overline{P} (Γ(S, O_{S})[[z]]). Taking s > 0 such that μ := x^{s} ∈ X_{∗}(T) ⊂ W̃ the assertion follows.

6 Central leaves and the product structure on Newton strata

Definition 6.1. Let 𝔽/𝔽_{q} be a field extension and let \underline{𝔾} be a local G-shtuka with K-structure over 𝔽. Let S be a noetherian 𝔽-scheme and let \underline{G} be a bounded local G-shtuka with K-structure over S. Consider the subset

C_{\underline{𝔾},S} = \{s ∈ S : \underline{𝔾}_{k} ≅ \underline{G}_{s} ⊗_{k(s)} k over an algebraically closed field k/k(s)\}.

By Corollary 6.7 below it is a locally closed subscheme. Any geometrically irreducible component of C_{\underline{𝔾},S} with the induced reduced subscheme structure is called a central leaf corresponding to \underline{𝔾} in S. Note that possibly the central leaves are only defined over a finite extension of 𝔽.

Lemma 6.2. Let 𝔽 ⊃ 𝔽_{q} be a field and let S be a reduced 𝔽-scheme. Let \underline{𝔾} = (K_{𝔽}, bσ^{∗}) and \underline{G} = (K_{S}, gσ^{∗}) be local G-shtukas with K-structure over 𝔽, respectively over S. Then the central leaf C_{\underline{𝔾},S} is constructible in S. In particular it contains a subset U ⊂ C_{\underline{𝔾},S} which is open and dense in the closure of C_{\underline{𝔾},S}.

Before proving this Lemma we provide two boundedness results.

Lemma 6.3. Let μ ∈ X_{∗}(T) be dominant. Then there is a positive integer e = e(μ) ∈ ℕ such that for every 𝔽_{q}-scheme S and for every h ∈ LG(S) which is bounded by μ

hI_{n+e}(S)h^{−1} ⊂ I_{n}(S) for all n .

Proof. It is enough to consider the case that h corresponds to the generic point of K_{0}z^{μ}K_{0} (as every h as in the assertion lies in the closure of that point). Especially S = Spec k′ for a field k′. We decompose h as g_{1}z^{μ}g_{2} for g_{1}, g_{2} ∈ K_{0}(k′). As K_{i} is a normal subgroup of K_{0} for each i we have hI_{n+e}(S)h^{−1} ⊂ hK_{n+e}(S)h^{−1} = g_{1}z^{μ}K_{n+e}(S)z^{−μ}g_{1}^{−1}. Choosing e so large that z^{μ}K_{n+e}z^{−μ} ⊂ K_{n+1} ⊂ I_{n} the lemma follows.

We call a subset of LG(k) = G(k((z))) bounded if it is contained in a finite union of K_{0}(k)- double cosets K_{0}(k)z^{μ}K_{0}(k) for dominant μ ∈ X_{∗}(T).

Lemma 6.4. Let B be a bounded subset of LG(k) for an algebraically closed field k and let K be an open subgroup of LG. Then there is a constant c = c(B, K) ∈ ℕ such that for each b ∈ B and each h ∈ I_{c}(k) there is an f ∈ K(k) with bh = f^{−1}bσ^{∗}(f).

Proof. In [HV, Theorem 10.1] a slightly stronger statement was proved for the case K = I_{n} for some n. As the subgroups I_{n} are cofinal in the set of open subgroups of LG, the lemma follows.

Proof of Lemma 6.2. Let μ be the Hodge point of b^{−1} and let e = e(μ) ∈ ℕ be the constant from Lemma 6.3. Also let B = {b} and let c = c(B, K) be the constant from Lemma 6.4. We consider the functor on the category of S-schemes X

Y (X) = { f : X → K/I_{c+e} : b_{X}^{−1}f^{−1}g_{X}σ^{∗}(f) ∈ I_{c}(X) }

Here the condition b_{X}^{−1}f^{−1}g_{X}σ^{∗}(f) ∈ I_{c}(X) means the following. Over an étale covering X̃ of X the morphism f is represented by an element f ∈ K(X̃). Then the condition is independent of the covering X̃ and the choice of the element f. Indeed if f = f̃h with h ∈ I_{c+e}(X̃) then

b−1 ; X̃f^{−1}gX̃σ^{∗}(f) = (b−1 ; X̃h^{−1}bX̃)b−1 ; X̃f̃^{−1}gX̃σ^{∗}(f̃)σ^{∗}(h) .

By Lemma 6.3 the terms b−1 ; X̃h^{−1}bX̃ and σ^{∗}(h) lie in I_{c}(X̃). This shows that Y is well defined and is representable by a scheme of finite type over S which is a closed subscheme of (K/I_{c+e})×_{𝔽_{q}} S. We claim that C_{\underline{𝔾},S} is the image of the morphism Y → S. From this the lemma follows by Chevalley’s theorem [EGA, IV_{1}, Corollaire 1.8.5] and [EGA, 0_{3}, Proposition 9.2.2].

To prove the claim let s ∈ C_{\underline{𝔾},S}, let k be an algebraically closed field, and let f_{s} ∈ K(k) be an isomorphism \underline{𝔾}_{k} ^{∼} ⟶ \underline{G}_{s} ⊗_{k(s)} k, that is f_{s}^{−1}g_{s}σ^{∗}(f_{s}) = b_{s}. Then in particular f_{s} ∈ Y (k) is a point mapping to s ∈ C_{\underline{𝔾},S}. Conversely let s ∈ S lie in the image of a point f_{s} ∈ Y (k) for an algebraically closed field k, that is f_{s} ∈ K(k) with b_{s}^{−1}f_{s}^{−1}g_{s}σ^{∗}(f_{s}) ∈ I_{c}(k). Then Lemma 6.4 yields an l ∈ K(k) with f_{s}^{−1}g_{s}σ^{∗}(f_{s}) = l^{−1}b_{s}σ^{∗}(l). Thus f_{s}l^{−1} is an isomorphism \underline{𝔾}_{s} ^{∼} ⟶ \underline{G}_{s} and so s belongs to C_{\underline{𝔾},S}.

To state the result on the product structure on Newton strata we need a generalization to more general K of affine Deligne-Lusztig varieties. Let b ∈ LG(k). Let μ ∈ X_{∗}(T) be dominant. Then the affine Deligne-Lusztig variety associated with (b, μ, K) is the reduced subscheme X_{μ,K}(b) of LG/K defined by

X_{μ,K}(b)(k) = {g ∈ LG/K(k) : g^{−1}bσ^{∗}(g) ∈ K_{0}(k)z^{μ}K_{0}(k)},  (6.1)

i.e. it is simply the reduced inverse image of X_{μ}(b) = X_{μ,K_{0}}(b) under LG/K → LG/K_{0}. The closed affine Deligne-Lusztig variety is X_{⪯μ,K}(b) = ⋃_{μ′⪯μ} X_{μ,K}(b). Note that one could replace the condition in (6.1) by any condition of the form g^{−1}bσ^{∗}(g) ∈ B for some bounded subset B of LG which a union of K-double cosets. In this way one obtains finer notions of affine Deligne-Lusztig varieties. Of particular interest is the case K = I where the set of double cosets is parametrized by W̃. For y ∈ W̃, the affine Deligne-Lusztig variety X_{y}(b) associated with (y, b) is defined as the reduced subscheme of the affine flag variety with

X_{y}(b)(k) = {g ∈ LG/I(k) : g^{−1}bσ^{∗}(g) ∈ I(k)yI(k)}.  (6.2)

Theorem 6.5. Let \underline{𝔾} = (K_{k}, bσ^{∗}) be a local G-shtuka with K-structure over an algebraically closed field k bounded by a dominant μ ∈ X_{∗}(T) with Newton point ν. Let N_{ν} be the Newton stratum of [b] in the universal deformation space of \underline{𝔾} bounded by μ (Proposition 3.6).

Let x be a P -fundamental alcove associated with [b] where the Levi subgroup M of P equals the centralizer of the M-dominant Newton point of x. Then there is a reduced scheme S and a finite surjective morphism S ↠ N_{ν} which factors into finite surjective morphisms

S →^{β} X_{⪯μ,K}(b)^{∧} ˆ×_{k} (I\overline{N}/x^{−1}I\overline{N}x)^{∧} →^{α} N_{ν}. Here X_{⪯μ,K}(b)^{∧} and (I\overline{N}/x^{−1}I\overline{N}x)^{∧} denote the

completions of X_{⪯μ,K}(b), respectively I\overline{N}/x^{−1}I\overline{N}x at 1. Furthermore, C_{\underline{𝔾},N_{ν}} is geometrically irreducible and equal to the image of \{1\}ˆ×_{k}(I\overline{N}/x^{−1}I\overline{N}x)^{∧} in N_{ν}.

In particular, X_{⪯μ,K}(b)^{∧} ˆ×_{k} (I\overline{N}/x^{−1}I\overline{N}x)^{∧} and N_{ν} are isomorphic up to finite surjective morphisms and there is a unique central leaf containing the special point of N_{ν}. Oort [Oo] proves a similar product decomposition of the Newton strata in the deformation space of a p-divisible group and in the Siegel moduli space. Theorem 6.5 is a generalization to arbitrary split reductive groups and level subgroups K of the translation of his result to the function field case.

Note that we do not claim that S or the morphisms occurring in the theorem are canonical. They will depend on various choices made in the course of the proof.

Proof. Step 1. Let g ∈ LG(k) with g^{−1}xσ^{∗}(g) = b. In order to make the proof easier to digest let us explain its structure. The key observation is the following. Let T be a k-scheme, let n be a positive integer and consider T -valued points h ∈ LG(T) and δ ∈ x^{−n}I\overline{N}(T)x^{n} ⊂ I(T) such that the induced morphism h : T → LG/K factors through X_{⪯μ,K}(b). Then we can construct out of (h, δ) the local G-shtuka (K_{T} , (gh)^{−1}xδσ^{∗}(gh)) which has two desired properties. Firstly, it has constant Newton point ν since all elements of xI(T) have constant Newton point ν by Remark 4.2 (b). Secondly, from h ∈ X_{⪯μ,K}(b)(T) we obtain that (gh)^{−1}xσ^{∗}(gh) = h^{−1}bσ^{∗}(h) is bounded by μ. If we choose n big enough, we may hope that σ^{∗}(gh)^{−1}δσ^{∗}(gh) ∈ K_{0}(T). Then (K_{T} , (gh)^{−1}xδσ^{∗}(gh)) will be bounded by μ. This already indicates how we will construct the morphism α : X_{⪯μ,K}(b)^{∧} ˆ×_{k} (I\overline{N}/x^{−1}I\overline{N}x)^{∧} → N_{ν}. The proof now proceeds in several steps. We construct S in step 2, estimate the n needed for the above in step 3, define the morphism α in step 4, the morphism β : S → X_{⪯μ,K}(b)^{∧} ˆ×_{k} (I\overline{N}/x^{−1}I\overline{N}x)^{∧} in steps 5 and 6, prove the finiteness and surjectivity of α and β in step 7, and compute the central leaf C_{\underline{𝔾},N_{ν}} in step 8. See the beginning of each step for more details.

Step 2. Construction of a first finite surjective morphism S → N_{ν} over which the universal local G-shtuka is isogenous to a completely slope divisible one, and computation of two bounds μ′ and μ′′. Consider the quasi-isogeny groups J_{b} of \underline{𝔾} and J_{x} of (K_{k}, xσ^{∗}). Then gJ_{b}g^{−1} = J_{x} ⊂ M(k((z))) for the Levi subgroup M by Corollary 3.4. Let μ′ be such that all elements of gjX_{⪯μ,K}(b)^{∧} are bounded by μ′ for all j ∈ J_{b} ∩ K. Such a μ′ exists by [HV, Lemma 3.13] on a quasi-compact connected neighborhood of 1 ∈ X_{⪯μ,K}(b) since X_{⪯μ,K}(b) is reduced, all gj have the same image in π_{1}(G), and one only has to consider a finite set of representatives j for (J_{b} ∩ K)/(J_{b} ∩ K ∩ g^{−1}Kg). Now let \underline{G} be the universal local G-shtuka over N_{ν}. We consider the normalization Ñ_{ν} → N_{ν} of N_{ν}. Since N_{ν} is excellent this is a finite morphism by [EGA, IV, Scholie 7.8.3(iii) and (vi)]. Hence Ñ_{ν} is normal and its connected components are integral by [EGA, II, Corollary 6.3.8]. Applying Corollary 4.15 to each such component we see that there is a finite surjective morphism S = Spec R_{S} → N_{ν} with R_{S} a direct sum of integral domains, such that over S the local G-shtuka \underline{G} is isomorphic to (K_{S}, l^{−1}xεσ^{∗}(l)σ^{∗}) for some l ∈ LG(S) and some ε ∈ I\overline{N}(S) whose reductions in the finitely many maximal ideals of R_{S} are g and 1, respectively. Since S is quasi-compact and reduced, and the image of l in π_{1}(G) is trivial, l is bounded by some dominant μ′′ according to [HV, Lemma 3.13].

Step 3. Computation of the bound on n needed in step 1, first replacement of S, and choice of a section (I\overline{N}/x^{−1}I\overline{N}x)^{∧} ≅ I_{x,n} ↪ x^{−n}I\overline{N}x^{n}. Let n_{K} ∈ ℕ with I_{n_{K}} ⊂ K. Let n ∈ ℕ with h^{−1}x^{−n}I\overline{N}x^{n}h ⊂ I_{n_{K}} for all h ∈ LG bounded by μ′ or by μ′′ (which exists by Remark 4.2 (e) and Lemma 6.3). By Lemma 4.3 there exists an n_{1} ≥ n with σ^{∗}(j̃)^{−1}x^{−n_{1}}I\overline{N}x^{n_{1}}σ^{∗}(j̃) ⊂ x^{−n}I\overline{N}x^{n} for all j̃ contained in the bounded set g(J_{b}∩K)g^{−1} ⊂ M(k((z))). By Lemma 4.11 we can replace S by a finite extension and multiply l by an element of I\overline{N}(S) to achieve that ε as above is even in x^{−n_{1}}I\overline{N}(S)x^{n_{1}} ⊂ x^{−n}I\overline{N}(S)x^{n}. Let I_{x,n} be the completion of (x^{−n}I\overline{N}x^{n})/(x^{−(n+1)}I\overline{N}x^{n+1}) at 1. Then I_{x,n} ≅ (I\overline{N}/x^{−1}I\overline{N}x)^{∧}. We choose a section I_{x,n} → x^{−n}I\overline{N}x^{n} using Lemma 2.1 and from now on identify I_{x,n} with its image. Note that as I\overline{N} is pro-unipotent we may lift \{1\} → x^{−(n+1)}I\overline{N}x^{n+1} (where 1 is the special point of I_{x,n} and the map is the restriction of the section) to a morphism I_{x,n} → x^{−(n+1)}I\overline{N}x^{n+1}. Multiplying the original section by the inverse of this map, we see that we may assume that the section maps the special point of I_{x,n} to 1 ∈ x^{−n}I\overline{N}x^{n}.

Step 4. Definition of α : X_{⪯μ,K}(b)^{∧} ˆ×_{k} I_{x,n} =: T̂ → N_{ν}. We define α(h, δ) := (gh)^{−1}xδσ^{∗}(gh) where h ∈ LG(T̂) is a chosen, fixed representative (existing by Lemma 2.3) of the universal element of X_{⪯μ,K}(b)(T̂) and where δ ∈ x^{−n}I\overline{N}(T̂)x^{n} is the image of the universal element. We consider the local G-shtuka (KT̂, (gh)^{−1}xδσ^{∗}(gh)σ^{∗}) over T̂ together with the isomorphism at its closed point h_{k} : (K_{k}, (gh)^{−1}xσ^{∗}(gh)σ^{∗}) ^{∼} ⟶ \underline{𝔾}. It yields a deformation of \underline{𝔾} over T̂ which is independent of the representative h up to canonical isomorphism. By the observation in step 1, this deformation has constant Newton point ν and is bounded by μ because the choice of n implies that σ^{∗}(gh)^{−1}δσ^{∗}(gh) ∈ K(T̂) ⊂ K_{0}(T̂). This indeed defines α : T̂ → N_{ν}.

Step 5. Final definition of S and construction of a morphism S → X_{⪯μ,K}(b)^{∧}, i.e. of the first component of β. We want to use the description of \underline{G}_{S} by l and ε to construct a finite surjective morphism β : S → X_{⪯μ,K}(b)^{∧} ˆ×_{k} I_{x,n}. We first define the map S → X_{⪯μ,K}(b)^{∧}. For this purpose we choose an r ∈ ℕ with K_{r} ⊂ K ∩ hKh^{−1} for all h ∈ X_{⪯μ,K}(b)^{∧} (use Lemma 6.3). In order to obtain the surjectivity of β in step 8 below we rename our S to S_{1} and replace it by the finite disjoint union ∐_{j∈Γ} S_{j} with S_{j} = S_{1} for all j ∈ Γ = (J_{b} ∩K)/(J_{b} ∩K_{r}). The first component of the morphism β is given on S_{j} by h = j^{−1}g^{−1}l ∈ (LG/K)(S_{j}). To show that h ∈ X_{⪯μ,K}(b)^{∧}(S_{j}) we consider it in every geometric generic point \overline{η} of S_{j}. By Corollary 4.12 there is a y ∈ x^{−n}I\overline{N}(\overline{η})x^{n} with xε = y^{−1}xσ^{∗}(y). Together with the choice of μ′′ and n this implies that h^{−1}bσ^{∗}(h) = f̃(l^{−1}xεσ^{∗}(l))σ^{∗}(f̃)^{−1} with f̃ = l^{−1}yl = (gjh)^{−1}ygjh ∈ K(η̄). Hence h\overline{η} ∈ X_{⪯μ,K}(b)(η̄). Since X_{⪯μ,K}(b) ⊂ LG/K is closed, S_{j} is reduced and complete, and the reduction of h in all closed points of S_{j} is 1 ∈ LG/K we see that h gives a morphism S_{j} → X_{⪯μ,K}(b)^{∧}.

Step 6. Definition of the second component S → I_{x,n} of β. On S_{j} we would like to choose δ ∈ I_{x,n}(S_{j}) equal to ε̃ := σ^{∗}(gj^{−1}g^{−1})εσ^{∗}(gjg^{−1}) ∈ x^{−n}I\overline{N}(S_{j})x^{n} because then (gh)^{−1}xε̃σ^{∗}(gh) = l^{−1}xεσ^{∗}(l) and αβ equals the projection S_{j} → N_{ν}. However, ε̃ is in general not in the image I_{x,n} of the chosen section, and needs first to be modified by σ-conjugating (gh)^{−1}xε̃σ^{∗}(gh) by further elements of K. To do that and to define the second component δ we use induction on m to define it modulo (σ^{∗})^{m}(𝔪) where 𝔪 denotes the maximal ideal of any component of R_{S_{j}}. More precisely, we want to show that for every m ∈ ℕ there is a g_{m} ∈ (gh)^{−1}x^{−n}I\overline{N}(S_{j})x^{n}gh such that g_{m}^{−1}(gh)^{−1}xε̃σ^{∗}(ghg_{m}) = (gh)^{−1}xε_{m}δ_{m}σ^{∗}(gh) for some ε_{m} ∈ x^{−(n+1)}I\overline{N}(S_{j})x^{n+1} which is congruent to 1 modulo (σ^{∗})^{m}(𝔪) and some δ_{m} ∈ I_{x,n}(S_{j}). In addition we choose g_{m} and δ_{m} in such a way that g_{m+1} ≡ g_{m} and δ_{m} ≡ δ_{m−1} modulo (σ^{∗})^{m}(𝔪). Note that our choice of μ′ and n implies that g_{m} ∈ (gh)^{−1}x^{−n}I\overline{N}x^{n}gh ⊂ K. Hence this iterative process of σ-conjugating (gh)^{−1}xε̃σ^{∗}(gh) only changes the trivialization of the local G-shtuka with K-structure. For m = 0 let g_{0} = 1, and let δ_{0} ∈ I_{x,n}(S_{j}) be the unique element with ε_{0} = ε̃δ_{0}^{−1} ∈ x^{−(n+1)}I\overline{N}(S_{j})x^{n+1}. That is, δ_{0}^{−1} is the image of ε̃^{−1} under the map

x^{−n}I\overline{N}x^{n} → x^{−n}I\overline{N}x^{n}/x^{−(n+1)}I\overline{N}x^{n+1} → I_{x,n}.  (6.3)

For the induction step we set g_{m+1} = g_{m}(gh)^{−1}xε_{m}x^{−1}gh. Then

g_{m+1}^{−1}(gh)^{−1}xε̃σ^{∗}(ghg_{m+1}) = ((gh)^{−1}xε_{m}x^{−1}gh)^{−1}(gh)^{−1}xε_{m}δ_{m}σ^{∗}(gh)σ^{∗}((gh)^{−1}xε_{m}x^{−1}gh)
= (gh)^{−1}xδ_{m}σ^{∗}(xε_{m}x^{−1})σ^{∗}(gh).

Note that σ^{∗}(xε_{m}x^{−1}) ∈ x^{−n}I\overline{N}(S_{j})x^{n} is congruent to 1 modulo (σ^{∗})^{m+1}(𝔪). We now decompose δ_{m}σ^{∗}(xε_{m}x^{−1}) as ε_{m+1}δ_{m+1} with δ_{m+1} ∈ I_{x,n}(S_{j}) and ε_{m+1} ∈ x^{−(n+1)}I\overline{N}(S_{j})x^{n+1} by taking δ_{m+1}^{−1} as the image of σ^{∗}(xε_{m}^{−1}x^{−1})δ_{m}^{−1} under the map (6.3). Then δ_{m+1} ≡ δ_{m} (mod (σ^{∗})^{m+1}(𝔪)) and ε_{m+1} ≡ 1 (mod (σ^{∗})^{m+1}(𝔪)). We define δ and ĝ to be the limits of the δ_{m}, respectively g_{m} and replace l by lĝ and h by hĝ. Note that this does not change the point h ∈ X_{⪯μ}(b)(S_{j})^{∧}. Then the Frobenius of the universal local G-shtuka over S_{j} is ĝ^{−1}l^{−1}xεσ^{∗}(lĝ) = (ghĝ)^{−1}xε̃σ^{∗}(ghĝ) = (gh)^{−1}xδσ^{∗}(gh). This defines the second component δ of β on S_{j} and we put everything together to obtain the morphism β : S = ∐_{j∈Γ} S_{j} → X_{⪯μ,K}(b)^{∧} ˆ×_{k} I_{x,n}, such that the projection S → N_{ν} factors through the morphisms β and α : X_{⪯μ,K}(b)^{∧} ˆ×_{k} I_{x,n} → N_{ν}.

Step 7. Proof that α and β are finite surjective. Clearly, the morphism β is finite and α is surjective because S → N_{ν} is. To show that also α is finite and β is surjective we write N_{ν} = Spec A, and X_{⪯μ,K}(b)^{∧} ˆ×_{k} I_{x,n} = Spec B, and S = Spec C. Since A is noetherian by Proposition 3.6, it suffices for the finiteness of α to show that the A-module B is via β^{∗} contained in the finite A-module C. Let 𝔟 = ker(B → C). Since B is reduced we only need to show that 𝔟 is contained in every minimal prime ideal 𝔭 of B. We will show that 𝔭 lies in the image of β. Then 𝔟 ⊂ 𝔭. Also since β is finite, hence closed, this will prove that β is surjective. Consider the morphism Spec C ⊗_{A} B/𝔭 → Spec B/𝔭 which is finite surjective by base change from Spec C → Spec A. Let 𝔓 ⊂ C ⊗_{A} B/𝔭 be a prime ideal mapping to 𝔭 and set R = (C ⊗_{A} B/𝔭)/𝔓. Then R is a complete local integral domain. It comes with two projections Spec R → Spec B, the first of which factors through Spec C, both mapping the maximal ideal of R to the closed point of Spec B. The projections yield two R-valued points (h, δ) and (h′, δ′) of X_{⪯μ,K}(b)^{∧} ˆ×_{k} I_{x,n} with reduction (1, 1), which have the same image in N_{ν}(R), i.e. (gh)^{−1}xδσ^{∗}(gh) = l̃^{−1}(gh′)^{−1}xδ′σ^{∗}(gh′)σ^{∗}(l̃) for the representatives h, h′ ∈ LG(R) induced by the choice of representative in the definition of α, and for some l̃∈ K(R) such that h′l̃h^{−1} has reduction 1. By construction the point (h, δ) lies in the image of β, and we have to show that (h′, δ′) likewise does.

To compare h, h′ we consider them in each geometric point of Spec R. We use again that over a perfect field xδ = y^{−1}xσ^{∗}(y) and xδ′ = (y′)^{−1}xσ^{∗}(y′) for some y, y′ ∈ x^{−n}I\overline{N}x^{n}. Together with the choice of n this implies that h^{−1}bσ^{∗}(h) = f̃^{−1}(h′)^{−1}bσ^{∗}(h′)σ^{∗}(f̃) for f̃ = (gh′)^{−1}y′(gh′)l̃(gh)^{−1}y^{−1}(gh) ∈ K, that is h′f̃h^{−1} ∈ J_{b}. Let ξ = h′h^{−1}. The computation just made shows that in every geometric point, ξ ∈ J_{b} · hKh^{−1}. We consider the image of ξ in LG/(hKh^{−1})(R), which lies in the image of J_{b} in this quotient (as this is the case in each geometric point and R is reduced). The latter is a discrete subscheme, its intersection with each bounded subscheme of LG/(hKh^{−1}) is finite. Hence the image of ξ is constant and equal to the image of some j ∈ J_{b}. Thus j^{−1}ξ ∈ hK(R)h^{−1} and h′ = jh as elements of LG/K(R). Considering the reduction in the special point of R we see that j ∈ K. Furthermore j is unique up to right multiplication by elements of J_{b} ∩ K ∩ hKh^{−1}. So by our choice of S = ∐_{j∈Γ} S_{j} with S_{j} = S_{1} we can consider the R-valued point Spec R → S which lies in some S_{j_{0}} also as a point in S_{jj_{0}}. As such it is mapped under β to a point (jh, δ̃) = (h′, δ̃) in X_{⪯μ,K}(b)^{∧} ˆ×_{k} I_{x,n}.

It thus remains to show that for h = h′ in LG/K(R), also δ = δ′ in I_{x,n}. Then (h′, δ′) lies in the image of β as desired. Let 𝔪_{R} be the maximal ideal of R. By definition δ, δ′ ∈ x^{−n}I\overline{N}x^{n}(R). Let m be minimal with δ′δ^{−1} ∉ x^{−(n+1)}I\overline{N}(R/(σ^{∗})^{m}(𝔪_{R}))x^{n+1}. As δ, δ′ are both in the image of the section of I_{x,n} this implies δ′δ^{−1} = 1 ∈ LG(R/(σ^{∗})^{m−1}(𝔪_{R})). Our assumption that the deformations associated with (h, δ) and (h, δ′) are isomorphic as deformations (i.e. via an isomorphism which is the identity on the special fiber) implies that there is an f ∈ LG(R) with f ≡ 1 (mod 𝔪_{R}) and f^{−1}xδσ^{∗}(f) = xδ′. This is equivalent to δ^{−1}x^{−1}fxδ′ = σ^{∗}(f). Considering the equation for f modulo (σ^{∗})^{l}(𝔪_{R}) for all l ∈ ℕ we see that f ∈ \overline{N}(R((z))) since x normalizes \overline{N}. Considering the equation modulo x^{−i}I\overline{N}x^{i} for the maximal i with f ∈ x^{−i}I\overline{N}x^{i}(R) we see that this i is at least n, especially x^{−1}f^{−1}x ∈ x^{−(n+1)}I\overline{N}x^{n+1}. Furthermore f = xδσ^{∗}(f)(δ′)^{−1}x^{−1} ≡ 1 (mod (σ^{∗})^{m−1}(𝔪_{R})) by induction, thus σ^{∗}(f) ≡ 1 (mod (σ^{∗})^{m}(𝔪_{R})). Hence over R/(σ^{∗})^{m}(𝔪_{R}) we have δ′δ^{−1} = δ′σ^{∗}(f)^{−1}δ^{−1} = x^{−1}f^{−1}x ∈ x^{−(n+1)}I\overline{N}(R/(σ^{∗})^{m}(𝔪_{R}))x^{n+1}. But this is a contradiction to our assumption δ′δ^{−1} ∉ x^{−(n+1)}I\overline{N}(R/(σ^{∗})^{m}(𝔪_{R}))x^{n+1}. This proves δ = δ′ from which the finiteness of α, the surjectivity of β, and hence the first assertion of the theorem follow.

Step 8. Computation of the central leaves. To compute C_{\underline{𝔾},N_{ν}} consider two geometric points of N_{ν} and some preimages (h, δ) and (h′, δ′) in X_{⪯μ,K}(b)^{∧} ˆ×_{k} I_{x,n}. The preceding calculation shows that the local G-shtukas with K-structure above these points can only belong to the same isomorphism class if h′ = jh for some j ∈ J_{b}. We are interested in the special case h′ = 1. We use that over a perfect field each element xδ or xδ′ with δ, δ′ ∈ x^{−n}I\overline{N}x^{n} is σ-conjugate by an element of x^{−n}I\overline{N}x^{n} to x. Thus the set of points (h, δ) ∈ X_{⪯μ,K}(b)^{∧} ˆ×_{k} I_{x,n} such that the local G-shtuka with K-structure corresponding to the image in N_{ν} is isomorphic to \underline{𝔾} consists of all pairs (j, δ) for some j ∈ (J_{b}/J_{b} ∩ K) ∩ X_{⪯μ,K}(b)^{∧} and arbitrary δ. As j ∈ X_{⪯μ,K}(b)^{∧}, it is bounded, and hence only a finite set of elements of J_{b}/J_{b} ∩ K can occur. Furthermore, an element of this finite set of closed points of LG/K only lies in X_{⪯μ,K}(b)^{∧} if it is equal to 1. Hence j = 1 and we see that C_{\underline{𝔾},N_{ν}} is of the form claimed in the theorem.

An analogous proof shows

Theorem 6.6. Let \underline{𝔾} = (I_{k}, bσ^{∗}) be a local G-shtuka with I-structure over k which is of affine Weyl type y and has Newton point ν. Let N_{ν} be the Newton stratum of b in the universal deformation space of \underline{𝔾} of Weyl type y. Let x be a P -fundamental alcove associated with the isogeny class of \underline{𝔾} where P is such that the Levi subgroup M of P equals the centralizer of the M-dominant Newton point of x. Then up to finite surjective morphisms, N_{ν} is isomorphic to the completion X_{y}(b)^{∧} ˆ×_{k}(I\overline{N}/x^{−1}I\overline{N}x)^{∧} of X_{y}(b) ×_{k} I\overline{N}/x^{−1}I\overline{N}x at (1, 1). Furthermore, C_{\underline{𝔾},N_{ν}} is geometrically irreducible and equal to the image of \{1\}ˆ×_{k}(I\overline{N}/x^{−1}I\overline{N}x)^{∧} in N_{ν}.

Corollary 6.7. Using the notation of Definition 6.1 let ν be the Newton point of \underline{𝔾} and let N_{ν,S} be the Newton stratum of ν in S. Then C_{\underline{𝔾},S} is closed in N_{ν,S}. In particular C_{\underline{𝔾},S} with the reduced structure is a locally closed subscheme of S.

Proof. We write \underline{𝔾} = (K_{k}, b_{0}σ^{∗}) for some b_{0} and let μ be such that b_{0} is bounded by μ. By Lemma 6.2 C_{\underline{𝔾},S} is constructible, so it is enough to show that it is stable under specialization in N_{ν,S}. Let \underline{F} = (K_{k}, bσ^{∗}) be the local G-shtuka with K-structure associated with a point y in the closure of C_{\underline{𝔾},S} and in N_{ν,S}. As the Newton points of \underline{𝔾} and \underline{F} coincide there is a g ∈ LG(k) whose equivalence class modulo K lies in X_{⪯μ,K}(b) with g^{−1}bσ^{∗}(g) = b_{0}, and all such g lie in one orbit of J = J_{b}. We may replace S by the completion of N_{ν,S} in y. Let N_{ν} be the closed Newton stratum in the universal deformation of \underline{F} bounded by μ. Then there is a morphism S → N_{ν} such that the local G-shtuka on S is the pull-back of the universal local G-shtuka with K-structure over N_{ν}. We consider the inverse image in X_{⪯μ,K}(b)^{∧} ˆ×_{k} (I\overline{N}/x^{−1}I\overline{N}x)^{∧} of the image of C_{\underline{𝔾},S} in N_{ν}. By assumption it is nonempty. We consider its first component (in X_{⪯μ,K}(b)^{∧}) in the different points of this inverse image. On the one hand, the local G-shtuka with K-structure corresponding to such a point is isomorphic to \underline{𝔾}, hence the first component has to be of the form jgK for some j ∈ J_{b}. Note that jgK ∈ LG/K is a closed point defined over k for each j. On the other hand it is in the completion of X_{⪯μ,K}(b) at 1. By the same argument as in the proof of the last assertion of Theorem 6.5 we see that the first component has to be constant, and equal to the unique closed point 1 of X_{⪯μ,K}(b)^{∧}. Thus K ∩ Jg ≠ ∅, hence b_{0} and b are σ-conjugate under K, and y ∈ C_{\underline{𝔾},S}.

Corollary 6.8. Let b ∈ LG(k) and let ν be the Newton point of b.

(a) Let μ ∈ X_{∗}(T) be dominant with X_{μ}(b) ≠ ∅. Then X_{⪯μ}(b) and X_{μ}(b) are equidimensional of dimension ⟨ρ, μ − ν⟩ − \frac{1}{2}(rk_{𝔽_{q}((z))}(G) − rk_{𝔽_{q}((z))}(J_{b})). In particular, X_{μ}(b) is open and dense in X_{⪯μ}(b).

(b) Let y ∈ W̃ with X_{y}(b) ≠ ∅. Let C be an irreducible component of X_{y}(b) and let ℓ(ν, y, C) denote the maximal length of a chain of Newton points ⪰ ν which is realizable in the universal deformation of Weyl type y of a point \underline{G} associated with g^{−1}bσ^{∗}(g) for some g in the smooth locus of C (see [HV, Definition 7.6]). Then dim C ≥ ℓ(y)−⟨2ρ, ν⟩−ℓ(ν, y). On the other hand dim X_{y}(b) = ℓ(y) − ⟨2ρ, ν⟩ − codim(N_{ν}) where codim(N_{ν}) denotes the codimension of the Newton stratum associated with ν in IyI.

Remark 6.9. For the affine Deligne-Lusztig varieties X_{μ}(b) it is known by [GHKR1], [Vi1] that their dimension is equal to ⟨ρ, μ−ν⟩− \frac{1}{2}(rk_{𝔽_{q}((z))}(G)−rk_{𝔽_{q}((z))}(J_{b})). In [HV] we proved both assertions of the corollary under the additional assumption that b is basic. Equidimensionality of X_{μ}(b) for b ∈ T (𝔽_{q}((z))) (not necessarily basic) is shown in [GHKR1, Proposition 2.17.1]. The last assertion of the corollary proves [Be, Conjecture 1]. In general, the affine Deligne-Lusztig varieties X_{y}(b) are not equidimensional (see [GH]).

Proof of Corollary 6.8. Let X be an irreducible component of the affine Deligne-Lusztig variety. Let g be a smooth point on X. By replacing b by g^{−1}bσ^{∗}(g) we may assume that g = 1 is smooth and lies on the component we want to consider. Let \underline{G} be the local G-shtuka with K_{0}- respectively I-structure associated with b. We consider its universal deformation by local G-shtukas bounded by μ, respectively of affine Weyl type y, which by Proposition 3.6 is prorepresentable by a complete noetherian local ring \overline{D} of dimension ⟨2ρ, μ⟩, respectively ℓ(y).

By [HV, Corollary 7.7], the codimension of an irreducible component of the Newton stratum associated with the Newton point ν of \underline{G} in Spec \overline{D} is less or equal to the maximal length of a chain of Newton points ordered by ⪯ and lying between μ and ν, respectively realizable in (IyI)_{b}^{∧} (in the second case this is what we call ℓ(ν, y) above). Thus in the first case we obtain as in [HV, Proposition 7.8] that the dimension of each irreducible component of the Newton stratum is at least ⟨ρ, μ + ν⟩ − \frac{1}{2}(rk_{𝔽_{q}((z))}(G) − rk_{𝔽_{q}((z))}(J_{b})). The assertions of the corollary now follow from the product structures in Theorem 6.5 and Theorem 6.6 together with the fact that dim(I\overline{N}/x^{−1}I\overline{N}x) = dim(I/(I ∩ x^{−1}Ix)) = ℓ(x) = ⟨2ρ, ν⟩ by [GHKR2, Proposition 13.1.3 (3)]. Here x is a P -fundamental alcove corresponding to [b].

Corollary 6.10. Let \underline{𝔾} be a local G-shtuka with K-structure over k bounded by some μ ∈ X_{∗}(T).

(a) Let ν be the Newton polygon of \underline{𝔾}. Then the closed Newton stratum in the universal deformation ring for deformations of \underline{𝔾} that are bounded by μ is equidimensional of dimension ⟨ρ, μ + ν⟩ − \frac{1}{2}(rk_{𝔽_{q}((z))}(G) − rk_{𝔽_{q}((z))}(J_{b})).

(b) The generic fiber of the universal local G-shtuka with K-structure over the universal deformation ring for deformations of \underline{𝔾} that are bounded by μ has Newton polygon μ.

Proof. The first assertion follows from Corollary 6.8 and Theorem 6.5. Let ν_{0} be the Newton point at the generic point of the deformation space. By [HV, Corollary 7.7(b)] we obtain

∑_{i}⌈⟨ω_{i}, μ − ν⟩⌉ ≤ ∑_{i}⌈⟨ω_{i}, ν_{0} − ν⟩⌉.

The two sides of this inequality are the maximal lengths of chains of comparable Newton polygons between ν and μ, respectively ν_{0} (see [Ch, Theorem 7.4 (iv)]). As ν_{0} ⪯ μ, the right hand side is strictly smaller than the left hand side unless μ = ν_{0}.

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Urs Hartl Universität Münster Mathematisches Institut Einsteinstr. 62 D – 48149 Münster Germany

www.math.uni-muenster.de/u/urs.hartl/

Eva Viehmann Universität Bonn Mathematisches Institut Endenicher Allee 60 D – 53115 Bonn Germany